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Properties

Label 2-40e2-80.67-c1-0-10
Degree $2$
Conductor $1600$
Sign $0.923 - 0.384i$
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  − 0.614i·3-s + (2.83 + 2.83i)7-s + 2.62·9-s + (−1.95 − 1.95i)11-s + 2.05·13-s + (4.06 + 4.06i)17-s + (−0.683 − 0.683i)19-s + (1.74 − 1.74i)21-s + (−4.95 + 4.95i)23-s − 3.45i·27-s + (−0.835 + 0.835i)29-s + 2.35i·31-s + (−1.20 + 1.20i)33-s + 4.54·37-s − 1.26i·39-s + ⋯
L(s)  = 1  − 0.354i·3-s + (1.07 + 1.07i)7-s + 0.874·9-s + (−0.590 − 0.590i)11-s + 0.569·13-s + (0.986 + 0.986i)17-s + (−0.156 − 0.156i)19-s + (0.380 − 0.380i)21-s + (−1.03 + 1.03i)23-s − 0.664i·27-s + (−0.155 + 0.155i)29-s + 0.423i·31-s + (−0.209 + 0.209i)33-s + 0.747·37-s − 0.202i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.923 - 0.384i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.078124391\)
\(L(\frac12)\) \(\approx\) \(2.078124391\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.614iT - 3T^{2} \)
7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 + 1.95i)T + 11iT^{2} \)
13 \( 1 - 2.05T + 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 + 0.683i)T + 19iT^{2} \)
23 \( 1 + (4.95 - 4.95i)T - 23iT^{2} \)
29 \( 1 + (0.835 - 0.835i)T - 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 + 0.849T + 43T^{2} \)
47 \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + (4.16 - 4.16i)T - 59iT^{2} \)
61 \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (4.39 + 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 2.75iT - 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + (-3.52 - 3.52i)T + 97iT^{2} \)
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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

−9.371964626107121975747310624980, −8.434335577998576631419563060615, −8.009972421257904484184942141522, −7.22473894177347957849827247372, −5.92662338464832281526589630112, −5.61452127065597075924681722310, −4.48120665470032193377043736593, −3.46339574987643401605486234446, −2.15942832295228110487589961611, −1.32227413455802908446336610594, 0.946980058591436350350036531080, 2.11528958732146277440306664712, 3.57133735775341517471236191478, 4.46562070749708771434936849958, 4.90748925540538364531407902974, 6.10587127632646430597475388511, 7.20144381910731617991037440465, 7.71888462740766258813300606286, 8.374104511660926073028402188019, 9.678142586482432467791861962476

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