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Properties

Label 2-40e2-80.69-c1-0-14
Degree $2$
Conductor $1600$
Sign $-0.611 + 0.791i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 − 2.32i)3-s − 0.982·7-s + 7.82i·9-s + (1.62 + 1.62i)11-s + (0.690 + 0.690i)13-s − 2.19i·17-s + (1.92 − 1.92i)19-s + (2.28 + 2.28i)21-s − 2.01·23-s + (11.2 − 11.2i)27-s + (5.27 − 5.27i)29-s − 0.435·31-s − 7.56i·33-s + (5.79 − 5.79i)37-s − 3.21i·39-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)3-s − 0.371·7-s + 2.60i·9-s + (0.490 + 0.490i)11-s + (0.191 + 0.191i)13-s − 0.532i·17-s + (0.441 − 0.441i)19-s + (0.498 + 0.498i)21-s − 0.420·23-s + (2.15 − 2.15i)27-s + (0.978 − 0.978i)29-s − 0.0781·31-s − 1.31i·33-s + (0.953 − 0.953i)37-s − 0.514i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7929824499\)
\(L(\frac12)\) \(\approx\) \(0.7929824499\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (2.32 + 2.32i)T + 3iT^{2} \)
7 \( 1 + 0.982T + 7T^{2} \)
11 \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \)
13 \( 1 + (-0.690 - 0.690i)T + 13iT^{2} \)
17 \( 1 + 2.19iT - 17T^{2} \)
19 \( 1 + (-1.92 + 1.92i)T - 19iT^{2} \)
23 \( 1 + 2.01T + 23T^{2} \)
29 \( 1 + (-5.27 + 5.27i)T - 29iT^{2} \)
31 \( 1 + 0.435T + 31T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (-0.507 + 0.507i)T - 43iT^{2} \)
47 \( 1 - 9.21iT - 47T^{2} \)
53 \( 1 + (-6.29 + 6.29i)T - 53iT^{2} \)
59 \( 1 + (5.67 + 5.67i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (4.53 + 4.53i)T + 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (0.683 + 0.683i)T + 83iT^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 - 5.54iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.186286363543980701858663970063, −7.933250142031371085590710441831, −7.43963715051181623729906855547, −6.48440456786597532427392435413, −6.20800342284619457095802545003, −5.15964137979473909809345372641, −4.34847635997398070296566746845, −2.71663017753002623587415125987, −1.60461400184367701837603505039, −0.46524389618395174938855984273, 1.04302368302139704603897760689, 3.16974783249683885640906316582, 3.91509178677265470247334143810, 4.73640992875833942150184093029, 5.65889805036442321002578407042, 6.16652864703437210705635103356, 6.96268439885519805808916442414, 8.341510912564338938811363874247, 9.129055977176409953302642348189, 9.892807268937840676552633467931

Graph of the $Z$-function along the critical line