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Properties

Label 2-40e2-80.29-c1-0-29
Degree $2$
Conductor $1600$
Sign $0.311 + 0.950i$
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  + (1.82 − 1.82i)3-s + 4.50·7-s − 3.68i·9-s + (1.64 − 1.64i)11-s + (−1.51 + 1.51i)13-s − 1.45i·17-s + (−2.67 − 2.67i)19-s + (8.24 − 8.24i)21-s − 2.37·23-s + (−1.24 − 1.24i)27-s + (−0.924 − 0.924i)29-s + 7.20·31-s − 5.99i·33-s + (5.21 + 5.21i)37-s + 5.55i·39-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)3-s + 1.70·7-s − 1.22i·9-s + (0.494 − 0.494i)11-s + (−0.421 + 0.421i)13-s − 0.353i·17-s + (−0.614 − 0.614i)19-s + (1.79 − 1.79i)21-s − 0.495·23-s + (−0.239 − 0.239i)27-s + (−0.171 − 0.171i)29-s + 1.29·31-s − 1.04i·33-s + (0.856 + 0.856i)37-s + 0.888i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.311 + 0.950i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.985165698\)
\(L(\frac12)\) \(\approx\) \(2.985165698\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.82i)T - 3iT^{2} \)
7 \( 1 - 4.50T + 7T^{2} \)
11 \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 + 1.45iT - 17T^{2} \)
19 \( 1 + (2.67 + 2.67i)T + 19iT^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 + (0.924 + 0.924i)T + 29iT^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + (-5.21 - 5.21i)T + 37iT^{2} \)
41 \( 1 - 6.41iT - 41T^{2} \)
43 \( 1 + (7.65 + 7.65i)T + 43iT^{2} \)
47 \( 1 + 2.51iT - 47T^{2} \)
53 \( 1 + (-1.50 - 1.50i)T + 53iT^{2} \)
59 \( 1 + (5.31 - 5.31i)T - 59iT^{2} \)
61 \( 1 + (1.02 + 1.02i)T + 61iT^{2} \)
67 \( 1 + (5.22 - 5.22i)T - 67iT^{2} \)
71 \( 1 + 1.92iT - 71T^{2} \)
73 \( 1 + 1.39T + 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 - 9.36iT - 89T^{2} \)
97 \( 1 + 18.6iT - 97T^{2} \)
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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

−8.844980673074747364240025953858, −8.390922528130398247430913932994, −7.80661785363953537166899745858, −7.05292962086423548901125860931, −6.24251181606647882306179115385, −4.97538955108546248642380736555, −4.21824235875508207530437558806, −2.87077757659965058319676517389, −2.02692769248363075315953762027, −1.15699635550395511304226366989, 1.62931195128120427108163033804, 2.56570467431833536323981472201, 3.80523701635103218721201595247, 4.45870011814310646038756826427, 5.08492589253836855482462755451, 6.26925894004411465117163057931, 7.62676649275313935214386211793, 8.086091382666655213215846662488, 8.723567026500229449660857616201, 9.532594541255629109398989015066

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