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Properties

Label 2-2301-2301.1910-c0-0-1
Degree $2$
Conductor $2301$
Sign $0.736 - 0.676i$
Analytic cond. $1.14834$
Root an. cond. $1.07161$
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  + (0.0695 + 0.424i)2-s + (−0.267 − 0.963i)3-s + (0.772 − 0.260i)4-s + (−0.771 + 1.13i)5-s + (0.390 − 0.180i)6-s + (0.365 + 0.689i)8-s + (−0.856 + 0.515i)9-s + (−0.536 − 0.248i)10-s + (−0.172 − 0.130i)11-s + (−0.457 − 0.674i)12-s + (0.856 + 0.515i)13-s + (1.30 + 0.439i)15-s + (0.381 − 0.290i)16-s + (−0.278 − 0.327i)18-s + (−0.299 + 1.08i)20-s + ⋯
L(s)  = 1  + (0.0695 + 0.424i)2-s + (−0.267 − 0.963i)3-s + (0.772 − 0.260i)4-s + (−0.771 + 1.13i)5-s + (0.390 − 0.180i)6-s + (0.365 + 0.689i)8-s + (−0.856 + 0.515i)9-s + (−0.536 − 0.248i)10-s + (−0.172 − 0.130i)11-s + (−0.457 − 0.674i)12-s + (0.856 + 0.515i)13-s + (1.30 + 0.439i)15-s + (0.381 − 0.290i)16-s + (−0.278 − 0.327i)18-s + (−0.299 + 1.08i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2301\)    =    \(3 \cdot 13 \cdot 59\)
Sign: $0.736 - 0.676i$
Analytic conductor: \(1.14834\)
Root analytic conductor: \(1.07161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2301} (1910, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2301,\ (\ :0),\ 0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.197386043\)
\(L(\frac12)\) \(\approx\) \(1.197386043\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.267 + 0.963i)T \)
13 \( 1 + (-0.856 - 0.515i)T \)
59 \( 1 + (-0.108 + 0.994i)T \)
good2 \( 1 + (-0.0695 - 0.424i)T + (-0.947 + 0.319i)T^{2} \)
5 \( 1 + (0.771 - 1.13i)T + (-0.370 - 0.928i)T^{2} \)
7 \( 1 + (-0.907 - 0.419i)T^{2} \)
11 \( 1 + (0.172 + 0.130i)T + (0.267 + 0.963i)T^{2} \)
17 \( 1 + (-0.907 + 0.419i)T^{2} \)
19 \( 1 + (-0.976 - 0.214i)T^{2} \)
23 \( 1 + (0.161 + 0.986i)T^{2} \)
29 \( 1 + (0.947 + 0.319i)T^{2} \)
31 \( 1 + (-0.976 + 0.214i)T^{2} \)
37 \( 1 + (0.561 + 0.827i)T^{2} \)
41 \( 1 + (-1.27 - 1.50i)T + (-0.161 + 0.986i)T^{2} \)
43 \( 1 + (1.15 - 0.878i)T + (0.267 - 0.963i)T^{2} \)
47 \( 1 + (-0.471 - 0.695i)T + (-0.370 + 0.928i)T^{2} \)
53 \( 1 + (-0.647 + 0.762i)T^{2} \)
61 \( 1 + (-0.315 - 1.92i)T + (-0.947 + 0.319i)T^{2} \)
67 \( 1 + (0.561 - 0.827i)T^{2} \)
71 \( 1 + (1.08 + 1.59i)T + (-0.370 + 0.928i)T^{2} \)
73 \( 1 + (0.994 + 0.108i)T^{2} \)
79 \( 1 + (-0.0865 + 0.311i)T + (-0.856 - 0.515i)T^{2} \)
83 \( 1 + (-1.34 + 1.27i)T + (0.0541 - 0.998i)T^{2} \)
89 \( 1 + (-0.246 + 1.50i)T + (-0.947 - 0.319i)T^{2} \)
97 \( 1 + (0.994 - 0.108i)T^{2} \)
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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

−9.066529952041814030968810021236, −8.039692120871483804155471707285, −7.62371111324114982304820552682, −6.93027547876588334807975211792, −6.30856114140965804737135794327, −5.83626349195240814616283180908, −4.58959049604703308673020881608, −3.28470876473869900413688057060, −2.56708514103062435696923209850, −1.40711506852313255546545918205, 0.903723323908947602646198992128, 2.43333244321860227836378884700, 3.69081540168216776847784400675, 3.95321705598583758717044480737, 5.06313317287568810823259231231, 5.73121311793543139121238558889, 6.76420198948592672515465383731, 7.74615324141828830179993285146, 8.469399983371868651072749894244, 9.031539229645502775814001849249

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