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Properties

Label 2-1265-1265.1077-c0-0-0
Degree $2$
Conductor $1265$
Sign $-0.277 - 0.960i$
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  + (−0.574 + 1.05i)3-s + (−0.755 − 0.654i)4-s + (0.540 + 0.841i)5-s + (−0.236 − 0.367i)9-s + (0.909 + 0.415i)11-s + (1.12 − 0.418i)12-s + (−1.19 + 0.0855i)15-s + (0.142 + 0.989i)16-s + (0.142 − 0.989i)20-s + (−0.909 − 0.415i)23-s + (−0.415 + 0.909i)25-s + (−0.673 + 0.0481i)27-s + (1.03 + 0.304i)31-s + (−0.959 + 0.718i)33-s + (−0.0621 + 0.432i)36-s + (−0.424 + 1.94i)37-s + ⋯
L(s)  = 1  + (−0.574 + 1.05i)3-s + (−0.755 − 0.654i)4-s + (0.540 + 0.841i)5-s + (−0.236 − 0.367i)9-s + (0.909 + 0.415i)11-s + (1.12 − 0.418i)12-s + (−1.19 + 0.0855i)15-s + (0.142 + 0.989i)16-s + (0.142 − 0.989i)20-s + (−0.909 − 0.415i)23-s + (−0.415 + 0.909i)25-s + (−0.673 + 0.0481i)27-s + (1.03 + 0.304i)31-s + (−0.959 + 0.718i)33-s + (−0.0621 + 0.432i)36-s + (−0.424 + 1.94i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ -0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7810291612\)
\(L(\frac12)\) \(\approx\) \(0.7810291612\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (-0.909 - 0.415i)T \)
23 \( 1 + (0.909 + 0.415i)T \)
good2 \( 1 + (0.755 + 0.654i)T^{2} \)
3 \( 1 + (0.574 - 1.05i)T + (-0.540 - 0.841i)T^{2} \)
7 \( 1 + (-0.281 - 0.959i)T^{2} \)
13 \( 1 + (0.281 - 0.959i)T^{2} \)
17 \( 1 + (-0.989 - 0.142i)T^{2} \)
19 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (-0.142 + 0.989i)T^{2} \)
31 \( 1 + (-1.03 - 0.304i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + (0.424 - 1.94i)T + (-0.909 - 0.415i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (-0.540 - 0.841i)T^{2} \)
47 \( 1 + (-0.300 - 0.300i)T + iT^{2} \)
53 \( 1 + (0.0855 - 0.114i)T + (-0.281 - 0.959i)T^{2} \)
59 \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (1.50 + 0.559i)T + (0.755 + 0.654i)T^{2} \)
71 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.989 - 0.142i)T^{2} \)
79 \( 1 + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.909 - 0.415i)T^{2} \)
89 \( 1 + (-1.89 + 0.557i)T + (0.841 - 0.540i)T^{2} \)
97 \( 1 + (-1.90 + 0.415i)T + (0.909 - 0.415i)T^{2} \)
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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

−10.27549054089823142884999302991, −9.528996865752880919393021474295, −8.900524011350013969733052017925, −7.67310803277663415959945980352, −6.36957344409347617515082409683, −6.04450348854120071255862325084, −4.86572242531054502298853701586, −4.38425795380137392588607639131, −3.28335358661697609256331622632, −1.68914521358438210380357112673, 0.789040340287500112945865925221, 2.02355256102648498403127785727, 3.64292456698243481090703837465, 4.51711094383003997358485923119, 5.62285996477719800667732970529, 6.20747242522042810958584430709, 7.24880548003956441808167162545, 7.996989627539828209108173237428, 8.866520334850076532747422100734, 9.371098065493502098374770966057

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