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Properties

Label 2-114-19.17-c1-0-1
Degree 22
Conductor 114114
Sign 0.877−0.479i0.877 - 0.479i
Analytic cond. 0.9102940.910294
Root an. cond. 0.9540930.954093
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (3.20 − 1.16i)5-s + (0.766 − 0.642i)6-s + (1.43 + 2.49i)7-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.592 + 3.35i)10-s + (0.173 − 0.300i)11-s + (0.499 + 0.866i)12-s + (−1.26 + 1.06i)13-s + (−2.70 + 0.984i)14-s + (−3.20 − 1.16i)15-s + (0.766 + 0.642i)16-s + (1.20 − 6.83i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (1.43 − 0.521i)5-s + (0.312 − 0.262i)6-s + (0.544 + 0.942i)7-s + (0.176 − 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.187 + 1.06i)10-s + (0.0523 − 0.0906i)11-s + (0.144 + 0.249i)12-s + (−0.351 + 0.294i)13-s + (−0.723 + 0.263i)14-s + (−0.827 − 0.301i)15-s + (0.191 + 0.160i)16-s + (0.292 − 1.65i)17-s + ⋯

Functional equation

Λ(s)=(114s/2 ΓC(s) L(s)=((0.877−0.479i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(114s/2 ΓC(s+1/2) L(s)=((0.877−0.479i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 114114    =    2⋅3⋅192 \cdot 3 \cdot 19
Sign: 0.877−0.479i0.877 - 0.479i
Analytic conductor: 0.9102940.910294
Root analytic conductor: 0.9540930.954093
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ114(55,⋅)\chi_{114} (55, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 114, ( :1/2), 0.877−0.479i)(2,\ 114,\ (\ :1/2),\ 0.877 - 0.479i)

Particular Values

L(1)L(1) ≈\approx 0.993270+0.253851i0.993270 + 0.253851i
L(12)L(\frac12) ≈\approx 0.993270+0.253851i0.993270 + 0.253851i
L(32)L(\frac{3}{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)
bad2 1+(0.173−0.984i)T 1 + (0.173 - 0.984i)T
3 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
19 1+(2.82−3.31i)T 1 + (2.82 - 3.31i)T
good5 1+(−3.20+1.16i)T+(3.83−3.21i)T2 1 + (-3.20 + 1.16i)T + (3.83 - 3.21i)T^{2}
7 1+(−1.43−2.49i)T+(−3.5+6.06i)T2 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2}
11 1+(−0.173+0.300i)T+(−5.5−9.52i)T2 1 + (-0.173 + 0.300i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.26−1.06i)T+(2.25−12.8i)T2 1 + (1.26 - 1.06i)T + (2.25 - 12.8i)T^{2}
17 1+(−1.20+6.83i)T+(−15.9−5.81i)T2 1 + (-1.20 + 6.83i)T + (-15.9 - 5.81i)T^{2}
23 1+(6.39+2.32i)T+(17.6+14.7i)T2 1 + (6.39 + 2.32i)T + (17.6 + 14.7i)T^{2}
29 1+(−1.10−6.25i)T+(−27.2+9.91i)T2 1 + (-1.10 - 6.25i)T + (-27.2 + 9.91i)T^{2}
31 1+(0.798+1.38i)T+(−15.5+26.8i)T2 1 + (0.798 + 1.38i)T + (-15.5 + 26.8i)T^{2}
37 1+11.2T+37T2 1 + 11.2T + 37T^{2}
41 1+(2.67+2.24i)T+(7.11+40.3i)T2 1 + (2.67 + 2.24i)T + (7.11 + 40.3i)T^{2}
43 1+(2.14−0.780i)T+(32.9−27.6i)T2 1 + (2.14 - 0.780i)T + (32.9 - 27.6i)T^{2}
47 1+(−0.971−5.51i)T+(−44.1+16.0i)T2 1 + (-0.971 - 5.51i)T + (-44.1 + 16.0i)T^{2}
53 1+(1.86+0.677i)T+(40.6+34.0i)T2 1 + (1.86 + 0.677i)T + (40.6 + 34.0i)T^{2}
59 1+(−0.0773+0.438i)T+(−55.4−20.1i)T2 1 + (-0.0773 + 0.438i)T + (-55.4 - 20.1i)T^{2}
61 1+(−11.7−4.28i)T+(46.7+39.2i)T2 1 + (-11.7 - 4.28i)T + (46.7 + 39.2i)T^{2}
67 1+(0.187+1.06i)T+(−62.9+22.9i)T2 1 + (0.187 + 1.06i)T + (-62.9 + 22.9i)T^{2}
71 1+(−15.6+5.68i)T+(54.3−45.6i)T2 1 + (-15.6 + 5.68i)T + (54.3 - 45.6i)T^{2}
73 1+(−9.51−7.98i)T+(12.6+71.8i)T2 1 + (-9.51 - 7.98i)T + (12.6 + 71.8i)T^{2}
79 1+(8.36+7.02i)T+(13.7+77.7i)T2 1 + (8.36 + 7.02i)T + (13.7 + 77.7i)T^{2}
83 1+(5.85+10.1i)T+(−41.5+71.8i)T2 1 + (5.85 + 10.1i)T + (-41.5 + 71.8i)T^{2}
89 1+(−1.37+1.15i)T+(15.4−87.6i)T2 1 + (-1.37 + 1.15i)T + (15.4 - 87.6i)T^{2}
97 1+(−0.634+3.59i)T+(−91.1−33.1i)T2 1 + (-0.634 + 3.59i)T + (-91.1 - 33.1i)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

−13.92656003282680194060307777169, −12.67851525005982149025215192136, −11.86632927489924931871864719651, −10.26685139703125150131090015462, −9.278447893386050270543240421544, −8.330520731403496127559970996531, −6.81491205309252606353575066094, −5.68367552208875440906799089145, −5.02212734313338549169827739806, −1.98266696372122383748274164302, 1.93162657894447532058412909292, 3.92441416291297868357236060348, 5.41114720546800750565035612468, 6.64844246915801982456791336779, 8.284280447930385401417654129636, 9.843652852307938666040817989673, 10.30234638788754787869128142978, 11.09990197580290045197938940752, 12.44666093164303591635835261242, 13.54599378759990768054358380766

Graph of the ZZ-function along the critical line