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Properties

Label 2-40e2-16.5-c1-0-17
Degree $2$
Conductor $1600$
Sign $0.989 - 0.146i$
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 + 4.50i·7-s − 3.68i·9-s + (1.64 + 1.64i)11-s + (−1.51 + 1.51i)13-s − 1.45·17-s + (2.67 − 2.67i)19-s + (8.24 + 8.24i)21-s + 2.37i·23-s + (−1.24 − 1.24i)27-s + (0.924 − 0.924i)29-s + 7.20·31-s + 5.99·33-s + (5.21 + 5.21i)37-s + 5.55i·39-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)3-s + 1.70i·7-s − 1.22i·9-s + (0.494 + 0.494i)11-s + (−0.421 + 0.421i)13-s − 0.353·17-s + (0.614 − 0.614i)19-s + (1.79 + 1.79i)21-s + 0.495i·23-s + (−0.239 − 0.239i)27-s + (0.171 − 0.171i)29-s + 1.29·31-s + 1.04·33-s + (0.856 + 0.856i)37-s + 0.888i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.989 - 0.146i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.989 - 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.491636826\)
\(L(\frac12)\) \(\approx\) \(2.491636826\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 + (-1.64 - 1.64i)T + 11iT^{2} \)
13 \( 1 + (1.51 - 1.51i)T - 13iT^{2} \)
17 \( 1 + 1.45T + 17T^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
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 \( 1 + 2.51T + 47T^{2} \)
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 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 - 9.36iT - 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−9.335251056055193661390798919094, −8.554234820416561961676130299558, −7.983584822541826323209695360401, −7.05135389464169674993346106755, −6.41789771626280861801135571852, −5.43017725706745723780046814780, −4.36794928842021878745559430980, −2.88828977101750678393801132279, −2.46155415813413943425417500095, −1.43552743855965804178304878874, 0.931615284290687550000716401203, 2.62937487655962655148999624991, 3.58361055983967991807042391665, 4.13578377446789482987942656849, 4.88613616822894105688721916879, 6.20509032540274006021198315689, 7.19994274986962679374464271620, 7.938347460610119656444874305451, 8.603109946750174421725091333621, 9.565881291396284397654838023223

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