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Properties

Label 2-40e2-16.13-c1-0-30
Degree $2$
Conductor $1600$
Sign $-0.743 + 0.668i$
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  + (−0.183 − 0.183i)3-s − 3.84i·7-s − 2.93i·9-s + (−1.60 + 1.60i)11-s + (1.80 + 1.80i)13-s + 4.93·17-s + (−4.77 − 4.77i)19-s + (−0.707 + 0.707i)21-s + 0.134i·23-s + (−1.09 + 1.09i)27-s + (−2.17 − 2.17i)29-s − 2.26·31-s + 0.588·33-s + (−4.35 + 4.35i)37-s − 0.664i·39-s + ⋯
L(s)  = 1  + (−0.106 − 0.106i)3-s − 1.45i·7-s − 0.977i·9-s + (−0.482 + 0.482i)11-s + (0.501 + 0.501i)13-s + 1.19·17-s + (−1.09 − 1.09i)19-s + (−0.154 + 0.154i)21-s + 0.0280i·23-s + (−0.209 + 0.209i)27-s + (−0.403 − 0.403i)29-s − 0.406·31-s + 0.102·33-s + (−0.715 + 0.715i)37-s − 0.106i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.743 + 0.668i$
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: \((2,\ 1600,\ (\ :1/2),\ -0.743 + 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082342236\)
\(L(\frac12)\) \(\approx\) \(1.082342236\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.183 + 0.183i)T + 3iT^{2} \)
7 \( 1 + 3.84iT - 7T^{2} \)
11 \( 1 + (1.60 - 1.60i)T - 11iT^{2} \)
13 \( 1 + (-1.80 - 1.80i)T + 13iT^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 + (4.77 + 4.77i)T + 19iT^{2} \)
23 \( 1 - 0.134iT - 23T^{2} \)
29 \( 1 + (2.17 + 2.17i)T + 29iT^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + (4.35 - 4.35i)T - 37iT^{2} \)
41 \( 1 + 3.34iT - 41T^{2} \)
43 \( 1 + (-2.70 + 2.70i)T - 43iT^{2} \)
47 \( 1 + 7.03T + 47T^{2} \)
53 \( 1 + (3.40 - 3.40i)T - 53iT^{2} \)
59 \( 1 + (0.107 - 0.107i)T - 59iT^{2} \)
61 \( 1 + (3.46 + 3.46i)T + 61iT^{2} \)
67 \( 1 + (-1.91 - 1.91i)T + 67iT^{2} \)
71 \( 1 + 9.32iT - 71T^{2} \)
73 \( 1 + 9.82iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (-8.80 - 8.80i)T + 83iT^{2} \)
89 \( 1 - 1.12iT - 89T^{2} \)
97 \( 1 + 6.10T + 97T^{2} \)
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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

−9.201837200895472199505662905079, −8.247293264037537570893146657108, −7.34039362860626734634787028436, −6.82297447146460503704566647698, −5.98849031964415597832231636375, −4.79988423598526255200371774448, −4.00712609587173946299241985795, −3.20482432092340129337173874822, −1.65097475132529507388118406639, −0.41884583728009316824912946815, 1.70585808061453445286504668740, 2.70956461859220300436811459789, 3.67745750125413632367679210622, 5.02561476304388160912839694024, 5.63476526659320350004501859014, 6.16116967433239793680877341920, 7.55100053271948578587046874306, 8.244793513331765298682173900762, 8.710962362962154756126898707041, 9.773278202334389392782161129066

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