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Properties

Label 2-40e2-80.29-c1-0-19
Degree $2$
Conductor $1600$
Sign $0.809 + 0.587i$
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  + (−1.03 + 1.03i)3-s − 1.49·7-s + 0.836i·9-s + (−0.423 + 0.423i)11-s + (1.85 − 1.85i)13-s − 6.50i·17-s + (−1.75 − 1.75i)19-s + (1.55 − 1.55i)21-s − 7.19·23-s + (−3.99 − 3.99i)27-s + (6.57 + 6.57i)29-s + 6.75·31-s − 0.880i·33-s + (−1.95 − 1.95i)37-s + 3.86i·39-s + ⋯
L(s)  = 1  + (−0.600 + 0.600i)3-s − 0.565·7-s + 0.278i·9-s + (−0.127 + 0.127i)11-s + (0.515 − 0.515i)13-s − 1.57i·17-s + (−0.403 − 0.403i)19-s + (0.339 − 0.339i)21-s − 1.49·23-s + (−0.767 − 0.767i)27-s + (1.22 + 1.22i)29-s + 1.21·31-s − 0.153i·33-s + (−0.321 − 0.321i)37-s + 0.618i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.809 + 0.587i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9656078795\)
\(L(\frac12)\) \(\approx\) \(0.9656078795\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 1.03i)T - 3iT^{2} \)
7 \( 1 + 1.49T + 7T^{2} \)
11 \( 1 + (0.423 - 0.423i)T - 11iT^{2} \)
13 \( 1 + (-1.85 + 1.85i)T - 13iT^{2} \)
17 \( 1 + 6.50iT - 17T^{2} \)
19 \( 1 + (1.75 + 1.75i)T + 19iT^{2} \)
23 \( 1 + 7.19T + 23T^{2} \)
29 \( 1 + (-6.57 - 6.57i)T + 29iT^{2} \)
31 \( 1 - 6.75T + 31T^{2} \)
37 \( 1 + (1.95 + 1.95i)T + 37iT^{2} \)
41 \( 1 + 7.70iT - 41T^{2} \)
43 \( 1 + (-6.13 - 6.13i)T + 43iT^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 + (-5.29 - 5.29i)T + 53iT^{2} \)
59 \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \)
61 \( 1 + (1.43 + 1.43i)T + 61iT^{2} \)
67 \( 1 + (-6.35 + 6.35i)T - 67iT^{2} \)
71 \( 1 + 4.08iT - 71T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-2.81 + 2.81i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 + 18.1iT - 97T^{2} \)
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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

−9.497560103275551999795788453756, −8.596313844197833055812114105257, −7.74495418902530325276926003749, −6.80247785437771447050750829108, −6.00140310666740887746681762373, −5.13836465895851421772406293648, −4.47471247071282279514288054302, −3.36061030976171426965699241792, −2.32508733939218450269067635436, −0.48565175931703581838337260126, 1.03943742894729288877853002110, 2.26242291459167113635955245455, 3.64869337250505797725764933512, 4.34089203402908759698261041614, 5.83040782321932593853891524779, 6.23263644176134846382847712282, 6.77188347686782552461262335613, 8.042521526570430120605038805538, 8.455200424740840639199790112108, 9.647027243468205740040215934894

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