Properties

Label 2-1265-1265.527-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.201 + 0.979i$
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  + (−0.114 − 0.0855i)3-s + (−0.909 + 0.415i)4-s + (−0.755 + 0.654i)5-s + (−0.275 − 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.139 + 0.0303i)12-s + (0.142 − 0.0101i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (−0.0987 + 0.264i)27-s + (0.0801 − 0.557i)31-s + (−0.0101 + 0.142i)33-s + (0.641 + 0.740i)36-s + (0.898 − 1.64i)37-s + ⋯
L(s)  = 1  + (−0.114 − 0.0855i)3-s + (−0.909 + 0.415i)4-s + (−0.755 + 0.654i)5-s + (−0.275 − 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.139 + 0.0303i)12-s + (0.142 − 0.0101i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (−0.0987 + 0.264i)27-s + (0.0801 − 0.557i)31-s + (−0.0101 + 0.142i)33-s + (0.641 + 0.740i)36-s + (0.898 − 1.64i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4926582363\)
\(L(\frac12)\) \(\approx\) \(0.4926582363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (0.540 + 0.841i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
good2 \( 1 + (0.909 - 0.415i)T^{2} \)
3 \( 1 + (0.114 + 0.0855i)T + (0.281 + 0.959i)T^{2} \)
7 \( 1 + (0.989 - 0.142i)T^{2} \)
13 \( 1 + (-0.989 - 0.142i)T^{2} \)
17 \( 1 + (0.755 - 0.654i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.0801 + 0.557i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (-0.898 + 1.64i)T + (-0.540 - 0.841i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.281 + 0.959i)T^{2} \)
47 \( 1 + (1.24 + 1.24i)T + iT^{2} \)
53 \( 1 + (-0.697 + 0.0498i)T + (0.989 - 0.142i)T^{2} \)
59 \( 1 + (1.37 - 1.19i)T + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (-1.94 + 0.424i)T + (0.909 - 0.415i)T^{2} \)
71 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (-0.755 - 0.654i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.540 - 0.841i)T^{2} \)
89 \( 1 + (0.215 + 1.49i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.459i)T + (0.540 - 0.841i)T^{2} \)
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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

−9.575014579219095114723421146297, −8.809682604487343869099502970311, −8.138319547427436238105054398152, −7.37107896896091002019015631645, −6.42063493646047922711969235514, −5.50183333163984470903916368031, −4.38992730005716411615374072053, −3.53870287791063571191273317644, −2.83951579124534615099245020820, −0.47288489317332877783183378660, 1.44386304369985993706027701968, 3.07083287344622959724550832747, 4.38258111430182724145904510961, 4.87890061456298884698386254265, 5.51539838499732834679518109833, 6.84403081458257562077730704041, 7.997818204272257664867789423794, 8.230887673709361572730830726409, 9.344018125845290092059404795699, 9.893170520138620857388475715201

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