Properties

Label 2-40e2-80.69-c1-0-4
Degree $2$
Conductor $1600$
Sign $0.345 - 0.938i$
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  + (−1.66 − 1.66i)3-s + 1.87·7-s + 2.56i·9-s + (3.29 + 3.29i)11-s + (1.90 + 1.90i)13-s − 2.57i·17-s + (−5.76 + 5.76i)19-s + (−3.12 − 3.12i)21-s − 7.58·23-s + (−0.728 + 0.728i)27-s + (−6.45 + 6.45i)29-s + 0.799·31-s − 10.9i·33-s + (2.69 − 2.69i)37-s − 6.33i·39-s + ⋯
L(s)  = 1  + (−0.962 − 0.962i)3-s + 0.708·7-s + 0.854i·9-s + (0.994 + 0.994i)11-s + (0.527 + 0.527i)13-s − 0.623i·17-s + (−1.32 + 1.32i)19-s + (−0.681 − 0.681i)21-s − 1.58·23-s + (−0.140 + 0.140i)27-s + (−1.19 + 1.19i)29-s + 0.143·31-s − 1.91i·33-s + (0.443 − 0.443i)37-s − 1.01i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.345 - 0.938i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8091881868\)
\(L(\frac12)\) \(\approx\) \(0.8091881868\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 1.66i)T + 3iT^{2} \)
7 \( 1 - 1.87T + 7T^{2} \)
11 \( 1 + (-3.29 - 3.29i)T + 11iT^{2} \)
13 \( 1 + (-1.90 - 1.90i)T + 13iT^{2} \)
17 \( 1 + 2.57iT - 17T^{2} \)
19 \( 1 + (5.76 - 5.76i)T - 19iT^{2} \)
23 \( 1 + 7.58T + 23T^{2} \)
29 \( 1 + (6.45 - 6.45i)T - 29iT^{2} \)
31 \( 1 - 0.799T + 31T^{2} \)
37 \( 1 + (-2.69 + 2.69i)T - 37iT^{2} \)
41 \( 1 - 0.946iT - 41T^{2} \)
43 \( 1 + (-0.829 + 0.829i)T - 43iT^{2} \)
47 \( 1 - 1.52iT - 47T^{2} \)
53 \( 1 + (6.97 - 6.97i)T - 53iT^{2} \)
59 \( 1 + (-6.84 - 6.84i)T + 59iT^{2} \)
61 \( 1 + (6.87 - 6.87i)T - 61iT^{2} \)
67 \( 1 + (3.73 + 3.73i)T + 67iT^{2} \)
71 \( 1 - 9.34iT - 71T^{2} \)
73 \( 1 - 0.886T + 73T^{2} \)
79 \( 1 - 3.07T + 79T^{2} \)
83 \( 1 + (-0.989 - 0.989i)T + 83iT^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 - 7.16iT - 97T^{2} \)
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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

−9.553236432000211224615578334155, −8.695065301266422096053601986362, −7.73554585315171776610500984914, −7.12170816709443803542777162181, −6.26856973869038694969590344966, −5.77970979177965966388642394413, −4.57756041302967336368557486606, −3.86360389783334692155531527158, −1.95283784222547811018843486380, −1.44310364564656867723493734719, 0.36132436477682076613212565828, 1.97035711402294519668305790717, 3.61496877809903355788017115950, 4.25373840209451551819782280521, 5.06923077076777487788811507324, 6.13176798359219293815331173243, 6.30177572024866556642731970576, 7.88098987207290392025861689735, 8.475216625836908210079979453630, 9.365283639606373801662447873200

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