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Properties

Label 2-1288-1288.1133-c0-0-1
Degree $2$
Conductor $1288$
Sign $0.854 + 0.519i$
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  + (−0.415 + 0.909i)2-s + (0.540 + 0.158i)3-s + (−0.654 − 0.755i)4-s + (−1.66 + 1.07i)5-s + (−0.368 + 0.425i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (−0.574 − 0.368i)9-s + (−0.281 − 1.95i)10-s + (−0.234 − 0.512i)12-s + (−0.258 − 1.80i)13-s + (0.841 + 0.540i)14-s + (−1.07 + 0.314i)15-s + (−0.142 + 0.989i)16-s + (0.574 − 0.368i)18-s + (0.708 + 0.817i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (0.540 + 0.158i)3-s + (−0.654 − 0.755i)4-s + (−1.66 + 1.07i)5-s + (−0.368 + 0.425i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (−0.574 − 0.368i)9-s + (−0.281 − 1.95i)10-s + (−0.234 − 0.512i)12-s + (−0.258 − 1.80i)13-s + (0.841 + 0.540i)14-s + (−1.07 + 0.314i)15-s + (−0.142 + 0.989i)16-s + (0.574 − 0.368i)18-s + (0.708 + 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1288\)    =    \(2^{3} \cdot 7 \cdot 23\)
Sign: $0.854 + 0.519i$
Analytic conductor: \(0.642795\)
Root analytic conductor: \(0.801745\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1288} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1288,\ (\ :0),\ 0.854 + 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4980010428\)
\(L(\frac12)\) \(\approx\) \(0.4980010428\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
good3 \( 1 + (-0.540 - 0.158i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (1.66 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.708 - 0.817i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.153 + 1.07i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (1.27 + 0.817i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−9.874604562321362093915358166086, −8.516701152942531919120736167936, −8.012106478606001111066237528285, −7.56659617640848812317812051218, −6.81397932383536033366855976185, −5.82646348442687194838774296234, −4.56536770201047493579122208289, −3.67235635075842235398027948282, −3.00798630074542437787058097321, −0.47965759687002671942515294071, 1.55110313413638505123345766043, 2.73253424583552652827390285046, 3.73501247393564020033931169177, 4.59391624043571693916983941810, 5.32113698752634877290942167963, 7.16017163164076475153317477838, 7.80405098951091873594295916600, 8.575530212130214642890949953541, 9.015074342985648547345491572682, 9.514007831143153347266236295264

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