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Properties

Label 2-40e2-40.37-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.945 + 0.326i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7419861790\)
\(L(\frac12)\) \(\approx\) \(0.7419861790\)
\(L(1)\) 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

−9.852846750045062303110477711126, −8.518131259915099924681028763603, −7.67641948739512780711892624946, −7.14669936269317272762810026517, −6.35404777405152463038642474250, −5.56915779428869497575206393200, −4.93844714229888410008396902722, −3.61498629578203627218172465436, −2.09494165064015195096684179690, −1.17002212834116417923128429689, 0.854792395949630825824052265300, 3.04929714164307614392700433941, 3.76692036608279088586439958730, 4.92850778866561085010260028843, 5.40027115487692109183849460081, 6.11209615441944494090003650544, 7.09959911852944383639502627036, 8.108862813009212543179465396233, 9.211079371241800975188443617768, 9.714720786236130314317281861911

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