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Properties

Label 2-1265-1265.1022-c0-0-0
Degree $2$
Conductor $1265$
Sign $-0.845 - 0.534i$
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  + (−1.71 + 0.373i)3-s + (−0.989 − 0.142i)4-s + (0.281 + 0.959i)5-s + (1.89 − 0.864i)9-s + (0.755 + 0.654i)11-s + (1.75 − 0.125i)12-s + (−0.841 − 1.54i)15-s + (0.959 + 0.281i)16-s + (−0.142 − 0.989i)20-s + (−0.654 + 0.755i)23-s + (−0.841 + 0.540i)25-s + (−1.51 + 1.13i)27-s + (−1.53 + 0.983i)31-s + (−1.54 − 0.841i)33-s + (−1.99 + 0.586i)36-s + (−1.12 − 0.418i)37-s + ⋯
L(s)  = 1  + (−1.71 + 0.373i)3-s + (−0.989 − 0.142i)4-s + (0.281 + 0.959i)5-s + (1.89 − 0.864i)9-s + (0.755 + 0.654i)11-s + (1.75 − 0.125i)12-s + (−0.841 − 1.54i)15-s + (0.959 + 0.281i)16-s + (−0.142 − 0.989i)20-s + (−0.654 + 0.755i)23-s + (−0.841 + 0.540i)25-s + (−1.51 + 1.13i)27-s + (−1.53 + 0.983i)31-s + (−1.54 − 0.841i)33-s + (−1.99 + 0.586i)36-s + (−1.12 − 0.418i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $-0.845 - 0.534i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ -0.845 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3353474003\)
\(L(\frac12)\) \(\approx\) \(0.3353474003\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.281 - 0.959i)T \)
11 \( 1 + (-0.755 - 0.654i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
good2 \( 1 + (0.989 + 0.142i)T^{2} \)
3 \( 1 + (1.71 - 0.373i)T + (0.909 - 0.415i)T^{2} \)
7 \( 1 + (-0.540 + 0.841i)T^{2} \)
13 \( 1 + (0.540 + 0.841i)T^{2} \)
17 \( 1 + (-0.281 - 0.959i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (1.53 - 0.983i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (1.12 + 0.418i)T + (0.755 + 0.654i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.909 - 0.415i)T^{2} \)
47 \( 1 + (0.494 + 0.494i)T + iT^{2} \)
53 \( 1 + (-0.767 - 1.40i)T + (-0.540 + 0.841i)T^{2} \)
59 \( 1 + (-0.557 - 1.89i)T + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (0.956 + 0.0683i)T + (0.989 + 0.142i)T^{2} \)
71 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.281 - 0.959i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.755 + 0.654i)T^{2} \)
89 \( 1 + (0.474 + 0.304i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.654 - 1.75i)T + (-0.755 + 0.654i)T^{2} \)
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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

−10.42999237922817493774694610962, −9.616925403989785500109458549142, −8.939571057545435216563819313933, −7.41171234368821343847559487424, −6.80682967591467789358140672782, −5.81033286311524390183086806660, −5.37402048404516482463786558583, −4.31459944354127628698329654078, −3.60734591062136921830258676921, −1.57176856165726017223677217405, 0.40200723529655240661595171774, 1.58898958651855492553034666464, 3.84515868532653192639978493108, 4.60556903407057465542870572120, 5.44455486357403873610471072769, 5.91996388509605666997792382450, 6.86576697296045580529640829290, 7.994685230919205378623807606071, 8.786115226446114184140150147612, 9.600681783025257312331497177025

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