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Properties

Label 2-40e2-80.67-c1-0-28
Degree $2$
Conductor $1600$
Sign $-0.412 + 0.910i$
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  + 1.54i·3-s + (−1.17 − 1.17i)7-s + 0.610·9-s + (−2.04 − 2.04i)11-s + 2.14·13-s + (−2.07 − 2.07i)17-s + (−4.47 − 4.47i)19-s + (1.81 − 1.81i)21-s + (−4.86 + 4.86i)23-s + 5.58i·27-s + (−5.51 + 5.51i)29-s − 5.72i·31-s + (3.16 − 3.16i)33-s − 11.0·37-s + 3.31i·39-s + ⋯
L(s)  = 1  + 0.892i·3-s + (−0.443 − 0.443i)7-s + 0.203·9-s + (−0.616 − 0.616i)11-s + 0.594·13-s + (−0.502 − 0.502i)17-s + (−1.02 − 1.02i)19-s + (0.396 − 0.396i)21-s + (−1.01 + 1.01i)23-s + 1.07i·27-s + (−1.02 + 1.02i)29-s − 1.02i·31-s + (0.550 − 0.550i)33-s − 1.81·37-s + 0.530i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.412 + 0.910i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.412 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4780251059\)
\(L(\frac12)\) \(\approx\) \(0.4780251059\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.54iT - 3T^{2} \)
7 \( 1 + (1.17 + 1.17i)T + 7iT^{2} \)
11 \( 1 + (2.04 + 2.04i)T + 11iT^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 + (2.07 + 2.07i)T + 17iT^{2} \)
19 \( 1 + (4.47 + 4.47i)T + 19iT^{2} \)
23 \( 1 + (4.86 - 4.86i)T - 23iT^{2} \)
29 \( 1 + (5.51 - 5.51i)T - 29iT^{2} \)
31 \( 1 + 5.72iT - 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 11.4iT - 41T^{2} \)
43 \( 1 + 0.251T + 43T^{2} \)
47 \( 1 + (-0.119 + 0.119i)T - 47iT^{2} \)
53 \( 1 - 2.69iT - 53T^{2} \)
59 \( 1 + (1.24 - 1.24i)T - 59iT^{2} \)
61 \( 1 + (2.48 + 2.48i)T + 61iT^{2} \)
67 \( 1 - 9.23T + 67T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (7.85 + 7.85i)T + 73iT^{2} \)
79 \( 1 + 4.86T + 79T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 + 3.63T + 89T^{2} \)
97 \( 1 + (-9.89 - 9.89i)T + 97iT^{2} \)
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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

−9.167476482905585712001928716150, −8.625938558619689240020870103375, −7.47811145195943780121953133102, −6.80638581957951175067527186376, −5.74070412287537824576563302096, −4.97038035412921321932040316102, −3.93968828468864265864183728014, −3.41915509767248816660099562637, −2.01400719255890316874997788531, −0.17243132231722709212456639619, 1.64571080528380369284450522890, 2.36707531216604610814280936688, 3.73017518874144691859959440133, 4.63465290859150852359717343601, 5.92204811110449802395944244129, 6.39786799693342517846622604765, 7.21218102934395435546018149818, 8.166822885191799019617049493553, 8.553568128514839934485696710387, 9.801891787650406569433372966274

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