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Properties

Label 2-1265-1265.1187-c0-0-0
Degree $2$
Conductor $1265$
Sign $0.0350 - 0.999i$
Analytic cond. $0.631317$
Root an. cond. $0.794554$
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  + (−1.86 − 0.133i)3-s + (0.540 + 0.841i)4-s + (−0.989 − 0.142i)5-s + (2.48 + 0.357i)9-s + (0.281 − 0.959i)11-s + (−0.898 − 1.64i)12-s + (1.83 + 0.398i)15-s + (−0.415 + 0.909i)16-s + (−0.415 − 0.909i)20-s + (−0.281 + 0.959i)23-s + (0.959 + 0.281i)25-s + (−2.76 − 0.602i)27-s + (−1.29 + 1.49i)31-s + (−0.654 + 1.75i)33-s + (1.04 + 2.28i)36-s + (1.17 + 1.56i)37-s + ⋯
L(s)  = 1  + (−1.86 − 0.133i)3-s + (0.540 + 0.841i)4-s + (−0.989 − 0.142i)5-s + (2.48 + 0.357i)9-s + (0.281 − 0.959i)11-s + (−0.898 − 1.64i)12-s + (1.83 + 0.398i)15-s + (−0.415 + 0.909i)16-s + (−0.415 − 0.909i)20-s + (−0.281 + 0.959i)23-s + (0.959 + 0.281i)25-s + (−2.76 − 0.602i)27-s + (−1.29 + 1.49i)31-s + (−0.654 + 1.75i)33-s + (1.04 + 2.28i)36-s + (1.17 + 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0350 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0350 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.0350 - 0.999i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.0350 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4508717245\)
\(L(\frac12)\) \(\approx\) \(0.4508717245\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.989 + 0.142i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
good2 \( 1 + (-0.540 - 0.841i)T^{2} \)
3 \( 1 + (1.86 + 0.133i)T + (0.989 + 0.142i)T^{2} \)
7 \( 1 + (0.755 - 0.654i)T^{2} \)
13 \( 1 + (-0.755 - 0.654i)T^{2} \)
17 \( 1 + (-0.909 + 0.415i)T^{2} \)
19 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.415 + 0.909i)T^{2} \)
31 \( 1 + (1.29 - 1.49i)T + (-0.142 - 0.989i)T^{2} \)
37 \( 1 + (-1.17 - 1.56i)T + (-0.281 + 0.959i)T^{2} \)
41 \( 1 + (0.959 - 0.281i)T^{2} \)
43 \( 1 + (0.989 + 0.142i)T^{2} \)
47 \( 1 + (-0.847 - 0.847i)T + iT^{2} \)
53 \( 1 + (0.398 - 0.148i)T + (0.755 - 0.654i)T^{2} \)
59 \( 1 + (-0.983 + 0.449i)T + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.334 + 0.613i)T + (-0.540 - 0.841i)T^{2} \)
71 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
73 \( 1 + (0.909 + 0.415i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (-0.281 + 0.959i)T^{2} \)
89 \( 1 + (-1.19 - 1.37i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (-1.28 - 0.959i)T + (0.281 + 0.959i)T^{2} \)
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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

−10.58044477841868199737043082987, −9.280288251899863071223172592078, −8.169976621830366479995686482248, −7.47717270947728388546919156165, −6.73865743532793529685918178691, −6.03617455799143923306594895902, −5.06450603628314098015937372779, −4.13116069995146217485599671453, −3.24710648075900842549481039691, −1.29610639852663413189511207186, 0.54647939050369131070633939717, 2.04625133138882173199972418923, 4.01687210214736779207292086815, 4.65606729606758472356011862611, 5.59013287977868420111145258984, 6.26862903018823360579574888547, 7.09581514178569684720010312524, 7.54969336323015514913970818220, 9.165048399360623283807373580359, 10.09008122962085432731855365370

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