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Properties

Label 2-40e2-40.3-c1-0-5
Degree $2$
Conductor $1600$
Sign $0.685 - 0.727i$
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  + (−1.41 − 1.41i)3-s + (−2.44 − 2.44i)7-s + 1.00i·9-s + 3.46·11-s + (−2.82 + 2.82i)13-s + (−4.89 + 4.89i)17-s + 3.46i·19-s + 6.92i·21-s + (2.44 − 2.44i)23-s + (−2.82 + 2.82i)27-s − 6.92·29-s − 8i·31-s + (−4.89 − 4.89i)33-s + (5.65 + 5.65i)37-s + 8.00·39-s + ⋯
L(s)  = 1  + (−0.816 − 0.816i)3-s + (−0.925 − 0.925i)7-s + 0.333i·9-s + 1.04·11-s + (−0.784 + 0.784i)13-s + (−1.18 + 1.18i)17-s + 0.794i·19-s + 1.51i·21-s + (0.510 − 0.510i)23-s + (−0.544 + 0.544i)27-s − 1.28·29-s − 1.43i·31-s + (−0.852 − 0.852i)33-s + (0.929 + 0.929i)37-s + 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5758476667\)
\(L(\frac12)\) \(\approx\) \(0.5758476667\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.41i)T + 3iT^{2} \)
7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + (2.82 - 2.82i)T - 13iT^{2} \)
17 \( 1 + (4.89 - 4.89i)T - 17iT^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (-2.44 + 2.44i)T - 23iT^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 + 8iT - 31T^{2} \)
37 \( 1 + (-5.65 - 5.65i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + 43iT^{2} \)
47 \( 1 + (2.44 + 2.44i)T + 47iT^{2} \)
53 \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 13.8iT - 61T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-4.89 - 4.89i)T + 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \)
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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

−9.561498834884044267050611606797, −8.807668132621285067339847620818, −7.58893855134780422336142818864, −6.92195232672530254256118620406, −6.40971938539000531718994220404, −5.79407672073832509243767362689, −4.27970850067461389088798801485, −3.83193802129051690171480337102, −2.20078449301024849869665512494, −1.02246506012590179625602511288, 0.29399751931933899239669375471, 2.34886821054014470265124995751, 3.30642491642968302049182804233, 4.47664779308938548116251163537, 5.18405141735942500293766934444, 5.91087484096098314056567389350, 6.75326870253452015757967440595, 7.54337788843499075482049281054, 9.056165926478938538236826590632, 9.235044177716358756879714047219

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