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Properties

Label 2-40e2-16.13-c1-0-5
Degree $2$
Conductor $1600$
Sign $0.813 - 0.581i$
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  + (−0.734 − 0.734i)3-s + 1.71i·7-s − 1.92i·9-s + (−2.82 + 2.82i)11-s + (−2.59 − 2.59i)13-s + 1.89·17-s + (2.89 + 2.89i)19-s + (1.25 − 1.25i)21-s − 2.00i·23-s + (−3.61 + 3.61i)27-s + (6.72 + 6.72i)29-s + 7.11·31-s + 4.15·33-s + (2.25 − 2.25i)37-s + 3.81i·39-s + ⋯
L(s)  = 1  + (−0.423 − 0.423i)3-s + 0.648i·7-s − 0.640i·9-s + (−0.852 + 0.852i)11-s + (−0.719 − 0.719i)13-s + 0.460·17-s + (0.664 + 0.664i)19-s + (0.274 − 0.274i)21-s − 0.418i·23-s + (−0.695 + 0.695i)27-s + (1.24 + 1.24i)29-s + 1.27·31-s + 0.722·33-s + (0.370 − 0.370i)37-s + 0.610i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181080385\)
\(L(\frac12)\) \(\approx\) \(1.181080385\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.734 + 0.734i)T + 3iT^{2} \)
7 \( 1 - 1.71iT - 7T^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
13 \( 1 + (2.59 + 2.59i)T + 13iT^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 + (-2.89 - 2.89i)T + 19iT^{2} \)
23 \( 1 + 2.00iT - 23T^{2} \)
29 \( 1 + (-6.72 - 6.72i)T + 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \)
41 \( 1 - 1.59iT - 41T^{2} \)
43 \( 1 + (8.06 - 8.06i)T - 43iT^{2} \)
47 \( 1 + 4.43T + 47T^{2} \)
53 \( 1 + (0.481 - 0.481i)T - 53iT^{2} \)
59 \( 1 + (-3.08 + 3.08i)T - 59iT^{2} \)
61 \( 1 + (-3.46 - 3.46i)T + 61iT^{2} \)
67 \( 1 + (-1.80 - 1.80i)T + 67iT^{2} \)
71 \( 1 + 0.379iT - 71T^{2} \)
73 \( 1 - 8.37iT - 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 6.50T + 97T^{2} \)
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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

−9.816511826942914324222656392475, −8.546379065155205967648784969981, −7.87978494409232328645027715507, −7.04432180966991974112300111403, −6.26948845169011756073238863729, −5.36475185629616669556877821068, −4.77166090598533462428259385738, −3.31442522153421690015045058852, −2.45900231975523151853405501910, −1.04197632753228201300202193829, 0.58605365306667816070841513759, 2.28365178352170160549711505823, 3.33246083100278702591001131430, 4.54388808982037225767074470539, 5.04356783779947803150320011821, 5.98319899577522965450256952121, 6.95336119848150592646878968424, 7.79469203307769198433834912505, 8.392825234384703016224129013780, 9.574432852604528688826923711707

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