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Properties

Label 2-40e2-16.5-c1-0-30
Degree $2$
Conductor $1600$
Sign $-0.942 + 0.335i$
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  + (0.0623 − 0.0623i)3-s − 0.375i·7-s + 2.99i·9-s + (−2.36 − 2.36i)11-s + (−1.76 + 1.76i)13-s − 4.64·17-s + (2.34 − 2.34i)19-s + (−0.0234 − 0.0234i)21-s − 2.07i·23-s + (0.373 + 0.373i)27-s + (2.55 − 2.55i)29-s − 8.51·31-s − 0.295·33-s + (−7.62 − 7.62i)37-s + 0.219i·39-s + ⋯
L(s)  = 1  + (0.0359 − 0.0359i)3-s − 0.142i·7-s + 0.997i·9-s + (−0.713 − 0.713i)11-s + (−0.489 + 0.489i)13-s − 1.12·17-s + (0.539 − 0.539i)19-s + (−0.00511 − 0.00511i)21-s − 0.433i·23-s + (0.0718 + 0.0718i)27-s + (0.474 − 0.474i)29-s − 1.52·31-s − 0.0513·33-s + (−1.25 − 1.25i)37-s + 0.0352i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.942 + 0.335i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.942 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2331099645\)
\(L(\frac12)\) \(\approx\) \(0.2331099645\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0623 + 0.0623i)T - 3iT^{2} \)
7 \( 1 + 0.375iT - 7T^{2} \)
11 \( 1 + (2.36 + 2.36i)T + 11iT^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
19 \( 1 + (-2.34 + 2.34i)T - 19iT^{2} \)
23 \( 1 + 2.07iT - 23T^{2} \)
29 \( 1 + (-2.55 + 2.55i)T - 29iT^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 + (7.62 + 7.62i)T + 37iT^{2} \)
41 \( 1 - 3.77iT - 41T^{2} \)
43 \( 1 + (6.21 + 6.21i)T + 43iT^{2} \)
47 \( 1 + 9.71T + 47T^{2} \)
53 \( 1 + (-3.03 - 3.03i)T + 53iT^{2} \)
59 \( 1 + (-8.11 - 8.11i)T + 59iT^{2} \)
61 \( 1 + (-0.728 + 0.728i)T - 61iT^{2} \)
67 \( 1 + (-0.969 + 0.969i)T - 67iT^{2} \)
71 \( 1 + 9.14iT - 71T^{2} \)
73 \( 1 + 7.56iT - 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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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

−8.909682857786406760566019438955, −8.351232791114934985884863122859, −7.37748543379199076209930424267, −6.83520076414966921868373108324, −5.60653966249655943241067929961, −4.98865466659256910943523190217, −4.02807349612893815883453756764, −2.78921273820607384694403996657, −1.93545950985304496711743980901, −0.082968800523666816844939253910, 1.66180220087171328902840062760, 2.87733635248160425599147246537, 3.77878343562670076811478655830, 4.90175501314760254159281948799, 5.57360610413495857230683735023, 6.71511147180162974560744181759, 7.23235209959025072587104524728, 8.264154551301105563589317505094, 8.953284119605245410623804641718, 9.879128394452624749248023685890

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