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Properties

Label 2-2513-2513.391-c0-0-0
Degree $2$
Conductor $2513$
Sign $-0.105 + 0.994i$
Analytic cond. $1.25415$
Root an. cond. $1.11988$
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  + (−0.0522 − 0.00645i)2-s + (−0.967 − 0.242i)4-s + (−0.352 − 0.935i)7-s + (0.0980 + 0.0378i)8-s + (−0.319 − 0.947i)9-s + (1.58 + 1.07i)11-s + (0.0123 + 0.0511i)14-s + (0.874 + 0.467i)16-s + (0.0105 + 0.0515i)18-s + (−0.0760 − 0.0666i)22-s + (−0.536 − 0.504i)23-s + (−0.996 + 0.0876i)25-s + (0.113 + 0.990i)28-s + (−0.449 − 1.61i)29-s + (−0.128 − 0.0906i)32-s + ⋯
L(s)  = 1  + (−0.0522 − 0.00645i)2-s + (−0.967 − 0.242i)4-s + (−0.352 − 0.935i)7-s + (0.0980 + 0.0378i)8-s + (−0.319 − 0.947i)9-s + (1.58 + 1.07i)11-s + (0.0123 + 0.0511i)14-s + (0.874 + 0.467i)16-s + (0.0105 + 0.0515i)18-s + (−0.0760 − 0.0666i)22-s + (−0.536 − 0.504i)23-s + (−0.996 + 0.0876i)25-s + (0.113 + 0.990i)28-s + (−0.449 − 1.61i)29-s + (−0.128 − 0.0906i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2513\)    =    \(7 \cdot 359\)
Sign: $-0.105 + 0.994i$
Analytic conductor: \(1.25415\)
Root analytic conductor: \(1.11988\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2513} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2513,\ (\ :0),\ -0.105 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7848121779\)
\(L(\frac12)\) \(\approx\) \(0.7848121779\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.352 + 0.935i)T \)
359 \( 1 + (-0.740 - 0.672i)T \)
good2 \( 1 + (0.0522 + 0.00645i)T + (0.969 + 0.243i)T^{2} \)
3 \( 1 + (0.319 + 0.947i)T^{2} \)
5 \( 1 + (0.996 - 0.0876i)T^{2} \)
11 \( 1 + (-1.58 - 1.07i)T + (0.368 + 0.929i)T^{2} \)
13 \( 1 + (-0.990 - 0.139i)T^{2} \)
17 \( 1 + (-0.960 + 0.277i)T^{2} \)
19 \( 1 + (0.992 + 0.122i)T^{2} \)
23 \( 1 + (0.536 + 0.504i)T + (0.0613 + 0.998i)T^{2} \)
29 \( 1 + (0.449 + 1.61i)T + (-0.855 + 0.517i)T^{2} \)
31 \( 1 + (-0.0263 - 0.999i)T^{2} \)
37 \( 1 + (-0.332 + 0.583i)T + (-0.510 - 0.860i)T^{2} \)
41 \( 1 + (-0.691 - 0.722i)T^{2} \)
43 \( 1 + (-1.24 - 0.267i)T + (0.912 + 0.408i)T^{2} \)
47 \( 1 + (-0.525 + 0.851i)T^{2} \)
53 \( 1 + (0.668 + 1.87i)T + (-0.774 + 0.632i)T^{2} \)
59 \( 1 + (0.00877 + 0.999i)T^{2} \)
61 \( 1 + (0.597 + 0.801i)T^{2} \)
67 \( 1 + (1.01 + 1.71i)T + (-0.479 + 0.877i)T^{2} \)
71 \( 1 + (-0.498 + 1.67i)T + (-0.836 - 0.547i)T^{2} \)
73 \( 1 + (0.00877 - 0.999i)T^{2} \)
79 \( 1 + (-0.331 + 0.0116i)T + (0.997 - 0.0701i)T^{2} \)
83 \( 1 + (-0.994 - 0.105i)T^{2} \)
89 \( 1 + (0.728 + 0.685i)T^{2} \)
97 \( 1 + (0.510 - 0.860i)T^{2} \)
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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

−9.310665252837854389004258546946, −8.142086302427495130149964041139, −7.41832982391650750884567837461, −6.41274234682876471697394501722, −6.03504167761967549792307430922, −4.67385462394140476874387208639, −4.02886840781946809693843373268, −3.58814618700358388633466791747, −1.85898737171439375682749295024, −0.59146821711594776754141000226, 1.43551551484903201511097732390, 2.82764291540499191630725520448, 3.68170365526582811178559952940, 4.47845437141529972040660019664, 5.65299575021277709284865516024, 5.85081623076977869686736000095, 7.07248425046210532903118708926, 8.032116365544233705204666442181, 8.662305783850110155006909258477, 9.149714188207029477008630026932

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