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Properties

Label 2-40e2-80.69-c1-0-17
Degree $2$
Conductor $1600$
Sign $0.639 - 0.768i$
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  + (1.42 + 1.42i)3-s + 0.690·7-s + 1.05i·9-s + (3.06 + 3.06i)11-s + (2.33 + 2.33i)13-s − 5.28i·17-s + (5.38 − 5.38i)19-s + (0.982 + 0.982i)21-s − 1.60·23-s + (2.77 − 2.77i)27-s + (−1.70 + 1.70i)29-s + 4.69·31-s + 8.71i·33-s + (−7.89 + 7.89i)37-s + 6.65i·39-s + ⋯
L(s)  = 1  + (0.821 + 0.821i)3-s + 0.261·7-s + 0.350i·9-s + (0.922 + 0.922i)11-s + (0.648 + 0.648i)13-s − 1.28i·17-s + (1.23 − 1.23i)19-s + (0.214 + 0.214i)21-s − 0.335·23-s + (0.533 − 0.533i)27-s + (−0.316 + 0.316i)29-s + 0.843·31-s + 1.51i·33-s + (−1.29 + 1.29i)37-s + 1.06i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.639 - 0.768i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.639 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.613222616\)
\(L(\frac12)\) \(\approx\) \(2.613222616\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 1.42i)T + 3iT^{2} \)
7 \( 1 - 0.690T + 7T^{2} \)
11 \( 1 + (-3.06 - 3.06i)T + 11iT^{2} \)
13 \( 1 + (-2.33 - 2.33i)T + 13iT^{2} \)
17 \( 1 + 5.28iT - 17T^{2} \)
19 \( 1 + (-5.38 + 5.38i)T - 19iT^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 + (1.70 - 1.70i)T - 29iT^{2} \)
31 \( 1 - 4.69T + 31T^{2} \)
37 \( 1 + (7.89 - 7.89i)T - 37iT^{2} \)
41 \( 1 - 5.49iT - 41T^{2} \)
43 \( 1 + (-0.256 + 0.256i)T - 43iT^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 + (4.99 - 4.99i)T - 53iT^{2} \)
59 \( 1 + (-1.46 - 1.46i)T + 59iT^{2} \)
61 \( 1 + (-9.33 + 9.33i)T - 61iT^{2} \)
67 \( 1 + (-1.94 - 1.94i)T + 67iT^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (-7.30 - 7.30i)T + 83iT^{2} \)
89 \( 1 - 1.81iT - 89T^{2} \)
97 \( 1 - 5.27iT - 97T^{2} \)
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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

−9.576289659484455615517912411485, −8.931222157216745250148363159368, −8.120660683426196648828572921796, −7.06826883480157415165688458605, −6.50880431945208661410592997086, −5.01960694791978588316248282647, −4.51940789170894052250237451654, −3.53455705711978975178899775606, −2.71523018488228636481360167769, −1.33187994376919008266842007004, 1.14189389585767239296114004315, 1.98544714685674611447673436379, 3.36546188016902862410239484724, 3.82428600688340494944532348070, 5.42172776219698096746178174348, 6.08138977070642287171851452296, 7.00660816017969674552209337280, 7.962042961103865397706680468688, 8.320090779955961651496392866289, 8.999806675748423530957036016883

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