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Properties

Label 2-40e2-40.13-c0-0-4
Degree $2$
Conductor $1600$
Sign $0.326 + 0.945i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s − 1.99i·9-s + i·11-s + (1.22 − 1.22i)17-s − 19-s + (−1.22 − 1.22i)27-s + (1.22 + 1.22i)33-s − 41-s + i·49-s − 2.99i·51-s + (−1.22 + 1.22i)57-s − 2·59-s + (1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − 0.999·81-s + ⋯
L(s)  = 1  + (1.22 − 1.22i)3-s − 1.99i·9-s + i·11-s + (1.22 − 1.22i)17-s − 19-s + (−1.22 − 1.22i)27-s + (1.22 + 1.22i)33-s − 41-s + i·49-s − 2.99i·51-s + (−1.22 + 1.22i)57-s − 2·59-s + (1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s − 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.326 + 0.945i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.326 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.652007706\)
\(L(\frac12)\) \(\approx\) \(1.652007706\)
\(L(1)\) 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 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.404626416132058791771185574201, −8.446974779000307566183543404653, −7.86454094459123523737352590448, −7.13706867704339582369871748700, −6.61808746312110430846732915380, −5.41196162029333550606420666417, −4.24248952825084641288886257553, −3.12359012735038097370924039648, −2.35064442244631507813249404902, −1.32656690544536693158679870940, 1.89992714791620668640772235271, 3.17141705400259499739940157965, 3.63717486889119172252718859434, 4.56025821499251120501190033874, 5.52988132017086683409340862898, 6.45335709132326780459928504062, 7.84930443752832033713611529931, 8.284311986780850397230206565043, 8.931992386524332664916810802988, 9.693304693790119304065521421403

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