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Properties

Label 2-1288-1288.909-c0-0-4
Degree $2$
Conductor $1288$
Sign $0.102 + 0.994i$
Analytic cond. $0.642795$
Root an. cond. $0.801745$
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  + (−0.841 + 0.540i)2-s + (0.281 − 1.95i)3-s + (0.415 − 0.909i)4-s + (1.45 − 0.425i)5-s + (0.822 + 1.80i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−2.80 − 0.822i)9-s + (−0.989 + 1.14i)10-s + (−1.66 − 1.07i)12-s + (0.708 − 0.817i)13-s + (−0.959 − 0.281i)14-s + (−0.425 − 2.96i)15-s + (−0.654 − 0.755i)16-s + (2.80 − 0.822i)18-s + (−0.234 + 0.512i)19-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.281 − 1.95i)3-s + (0.415 − 0.909i)4-s + (1.45 − 0.425i)5-s + (0.822 + 1.80i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−2.80 − 0.822i)9-s + (−0.989 + 1.14i)10-s + (−1.66 − 1.07i)12-s + (0.708 − 0.817i)13-s + (−0.959 − 0.281i)14-s + (−0.425 − 2.96i)15-s + (−0.654 − 0.755i)16-s + (2.80 − 0.822i)18-s + (−0.234 + 0.512i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.102 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.102 + 0.994i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.102 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034923473\)
\(L(\frac12)\) \(\approx\) \(1.034923473\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
good3 \( 1 + (-0.281 + 1.95i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (-1.45 + 0.425i)T + (0.841 - 0.540i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.234 - 0.512i)T + (-0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 + (-0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.368 - 0.425i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.959 + 0.281i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (1.41 - 0.909i)T + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.544 + 0.627i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−9.253694915235597584281683858179, −8.656681391336633095134533289029, −8.096425559663205070479384856772, −7.37867545484393755118306456681, −6.29267833797921328354386990129, −5.85519058825142750801069841761, −5.35970074055665400690880113598, −2.76305004045995491009091094457, −1.85037519932057185299678792105, −1.30576748683286032218510630493, 1.92172441887968405579905805951, 2.89775109710405501658231046839, 3.96242583350213411657863154099, 4.64698180340078025070202021638, 5.82268996244107455242860225757, 6.74001188784640858038115565790, 8.106966850888070320969770171896, 8.816771609181013249063809482862, 9.447746151097676691405214603943, 10.03844878948247812622668483756

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