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Properties

Label 2-1450-5.4-c1-0-30
Degree 22
Conductor 14501450
Sign −0.894+0.447i-0.894 + 0.447i
Analytic cond. 11.578311.5783
Root an. cond. 3.402693.40269
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  − i·2-s − i·3-s − 4-s − 6-s + 2i·7-s + i·8-s + 2·9-s − 3·11-s + i·12-s − i·13-s + 2·14-s + 16-s − 8i·17-s − 2i·18-s + 2·21-s + 3i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353i·8-s + 0.666·9-s − 0.904·11-s + 0.288i·12-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s − 1.94i·17-s − 0.471i·18-s + 0.436·21-s + 0.639i·22-s + ⋯

Functional equation

Λ(s)=(1450s/2 ΓC(s) L(s)=((−0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1450s/2 ΓC(s+1/2) L(s)=((−0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14501450    =    2⋅52⋅292 \cdot 5^{2} \cdot 29
Sign: −0.894+0.447i-0.894 + 0.447i
Analytic conductor: 11.578311.5783
Root analytic conductor: 3.402693.40269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1450(349,⋅)\chi_{1450} (349, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1450, ( :1/2), −0.894+0.447i)(2,\ 1450,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.2228118541.222811854
L(12)L(\frac12) ≈\approx 1.2228118541.222811854
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+iT 1 + iT
5 1 1
29 1−T 1 - T
good3 1+iT−3T2 1 + iT - 3T^{2}
7 1−2iT−7T2 1 - 2iT - 7T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1+iT−13T2 1 + iT - 13T^{2}
17 1+8iT−17T2 1 + 8iT - 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1−4iT−23T2 1 - 4iT - 23T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+11iT−43T2 1 + 11iT - 43T^{2}
47 1+13iT−47T2 1 + 13iT - 47T^{2}
53 1+11iT−53T2 1 + 11iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1−12iT−67T2 1 - 12iT - 67T^{2}
71 1−2T+71T2 1 - 2T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1+15T+79T2 1 + 15T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−2iT−97T2 1 - 2iT - 97T^{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

−9.282708863658135757727897311368, −8.487515671598840071247803635051, −7.48402816835388469244948131603, −7.01138723621797215961796653114, −5.56138675528115967421074827793, −5.14278588832138914268892894165, −3.86934664072090993796106148388, −2.72117007321105440843291562410, −1.99883379181252830533277575355, −0.50934430537415501617057669376, 1.43337419471930793188795317925, 3.13815152938881594795222410751, 4.31172238911361323212801375920, 4.58615571580410205780919613967, 5.87624576083343006235451222415, 6.55377279749605764139928304380, 7.57394862267748566163786555293, 8.074362155823193822037270074564, 9.041531891321149447817572604306, 9.885830763888100270512883690511

Graph of the ZZ-function along the critical line