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Properties

Label 2-175-175.103-c3-0-20
Degree 22
Conductor 175175
Sign −0.155−0.987i-0.155 - 0.987i
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 − 0.256i)2-s + (5.67 + 7.00i)3-s + (15.9 + 1.67i)4-s + (−11.0 + 1.57i)5-s + (−25.9 − 35.7i)6-s + (17.8 − 5.00i)7-s + (−38.9 − 6.16i)8-s + (−11.2 + 52.9i)9-s + (54.6 − 4.85i)10-s + (46.2 − 9.84i)11-s + (78.7 + 121. i)12-s + (77.7 + 39.5i)13-s + (−88.6 + 19.9i)14-s + (−73.7 − 68.5i)15-s + (63.6 + 13.5i)16-s + (−1.45 − 3.79i)17-s + ⋯
L(s)  = 1  + (−1.73 − 0.0907i)2-s + (1.09 + 1.34i)3-s + (1.99 + 0.209i)4-s + (−0.990 + 0.140i)5-s + (−1.76 − 2.43i)6-s + (0.962 − 0.270i)7-s + (−1.72 − 0.272i)8-s + (−0.417 + 1.96i)9-s + (1.72 − 0.153i)10-s + (1.26 − 0.269i)11-s + (1.89 + 2.91i)12-s + (1.65 + 0.844i)13-s + (−1.69 + 0.380i)14-s + (−1.26 − 1.18i)15-s + (0.994 + 0.211i)16-s + (−0.0207 − 0.0541i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.155−0.987i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.155−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.155−0.987i-0.155 - 0.987i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), −0.155−0.987i)(2,\ 175,\ (\ :3/2),\ -0.155 - 0.987i)

Particular Values

L(2)L(2) ≈\approx 0.729415+0.853541i0.729415 + 0.853541i
L(12)L(\frac12) ≈\approx 0.729415+0.853541i0.729415 + 0.853541i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(11.0−1.57i)T 1 + (11.0 - 1.57i)T
7 1+(−17.8+5.00i)T 1 + (-17.8 + 5.00i)T
good2 1+(4.89+0.256i)T+(7.95+0.836i)T2 1 + (4.89 + 0.256i)T + (7.95 + 0.836i)T^{2}
3 1+(−5.67−7.00i)T+(−5.61+26.4i)T2 1 + (-5.67 - 7.00i)T + (-5.61 + 26.4i)T^{2}
11 1+(−46.2+9.84i)T+(1.21e3−541.i)T2 1 + (-46.2 + 9.84i)T + (1.21e3 - 541. i)T^{2}
13 1+(−77.7−39.5i)T+(1.29e3+1.77e3i)T2 1 + (-77.7 - 39.5i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(1.45+3.79i)T+(−3.65e3+3.28e3i)T2 1 + (1.45 + 3.79i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−3.63−34.5i)T+(−6.70e3+1.42e3i)T2 1 + (-3.63 - 34.5i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−7.51+143.i)T+(−1.21e4−1.27e3i)T2 1 + (-7.51 + 143. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(107.−147.i)T+(−7.53e3−2.31e4i)T2 1 + (107. - 147. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(58.5−131.i)T+(−1.99e4−2.21e4i)T2 1 + (58.5 - 131. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(72.6−47.1i)T+(2.06e4−4.62e4i)T2 1 + (72.6 - 47.1i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(30.3−9.86i)T+(5.57e4−4.05e4i)T2 1 + (30.3 - 9.86i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(68.8+68.8i)T+7.95e4iT2 1 + (68.8 + 68.8i)T + 7.95e4iT^{2}
47 1+(198.+76.3i)T+(7.71e4+6.94e4i)T2 1 + (198. + 76.3i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(92.6−75.0i)T+(3.09e4−1.45e5i)T2 1 + (92.6 - 75.0i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(403.−448.i)T+(−2.14e4−2.04e5i)T2 1 + (403. - 448. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(315.−284.i)T+(2.37e4−2.25e5i)T2 1 + (315. - 284. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−301.+115.i)T+(2.23e5−2.01e5i)T2 1 + (-301. + 115. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−471.−342.i)T+(1.10e5+3.40e5i)T2 1 + (-471. - 342. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−216.+333.i)T+(−1.58e5−3.55e5i)T2 1 + (-216. + 333. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(411.+925.i)T+(−3.29e5+3.66e5i)T2 1 + (411. + 925. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(73.2−462.i)T+(−5.43e5−1.76e5i)T2 1 + (73.2 - 462. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−328.−364.i)T+(−7.36e4+7.01e5i)T2 1 + (-328. - 364. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(−83.1−524.i)T+(−8.68e5+2.82e5i)T2 1 + (-83.1 - 524. i)T + (-8.68e5 + 2.82e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.80578604442349709788429413218, −10.97558977889748539371305861793, −10.52537121375023669248661143426, −9.086727990714108234426655248520, −8.728404723691379224173631853047, −8.051507080382249264236091742162, −6.79453793544935949794940451257, −4.36595316285152382771143626458, −3.42602419950622478736260337655, −1.50671022065758803929657938531, 0.932390455863758677552950238835, 1.79451606203804622517188300616, 3.55153968590262628846538162774, 6.31046082162699717330461564617, 7.43820277285533410402696230501, 7.988535359315447014482150460998, 8.647155418130342840488174997367, 9.368322944722939701654198024688, 11.21452305077957462067027484365, 11.57390747946365004935016644545

Graph of the ZZ-function along the critical line