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Properties

Label 2-2301-2301.1520-c0-0-1
Degree $2$
Conductor $2301$
Sign $0.826 + 0.562i$
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  + (−1.62 + 0.977i)2-s + (−0.370 − 0.928i)3-s + (1.21 − 2.29i)4-s + (1.80 − 0.196i)5-s + (1.50 + 1.14i)6-s + (0.163 + 3.01i)8-s + (−0.725 + 0.687i)9-s + (−2.73 + 2.08i)10-s + (−0.181 + 0.267i)11-s + (−2.57 − 0.280i)12-s + (−0.725 − 0.687i)13-s + (−0.850 − 1.60i)15-s + (−1.75 − 2.58i)16-s + (0.507 − 1.82i)18-s + (1.74 − 4.37i)20-s + ⋯
L(s)  = 1  + (−1.62 + 0.977i)2-s + (−0.370 − 0.928i)3-s + (1.21 − 2.29i)4-s + (1.80 − 0.196i)5-s + (1.50 + 1.14i)6-s + (0.163 + 3.01i)8-s + (−0.725 + 0.687i)9-s + (−2.73 + 2.08i)10-s + (−0.181 + 0.267i)11-s + (−2.57 − 0.280i)12-s + (−0.725 − 0.687i)13-s + (−0.850 − 1.60i)15-s + (−1.75 − 2.58i)16-s + (0.507 − 1.82i)18-s + (1.74 − 4.37i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2301\)    =    \(3 \cdot 13 \cdot 59\)
Sign: $0.826 + 0.562i$
Analytic conductor: \(1.14834\)
Root analytic conductor: \(1.07161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2301} (1520, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2301,\ (\ :0),\ 0.826 + 0.562i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6153999038\)
\(L(\frac12)\) \(\approx\) \(0.6153999038\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.370 + 0.928i)T \)
13 \( 1 + (0.725 + 0.687i)T \)
59 \( 1 + (-0.161 - 0.986i)T \)
good2 \( 1 + (1.62 - 0.977i)T + (0.468 - 0.883i)T^{2} \)
5 \( 1 + (-1.80 + 0.196i)T + (0.976 - 0.214i)T^{2} \)
7 \( 1 + (-0.796 + 0.605i)T^{2} \)
11 \( 1 + (0.181 - 0.267i)T + (-0.370 - 0.928i)T^{2} \)
17 \( 1 + (-0.796 - 0.605i)T^{2} \)
19 \( 1 + (0.947 - 0.319i)T^{2} \)
23 \( 1 + (0.856 - 0.515i)T^{2} \)
29 \( 1 + (-0.468 - 0.883i)T^{2} \)
31 \( 1 + (0.947 + 0.319i)T^{2} \)
37 \( 1 + (0.994 + 0.108i)T^{2} \)
41 \( 1 + (-0.458 + 1.65i)T + (-0.856 - 0.515i)T^{2} \)
43 \( 1 + (1.01 + 1.50i)T + (-0.370 + 0.928i)T^{2} \)
47 \( 1 + (-1.58 - 0.172i)T + (0.976 + 0.214i)T^{2} \)
53 \( 1 + (-0.267 - 0.963i)T^{2} \)
61 \( 1 + (-1.62 + 0.977i)T + (0.468 - 0.883i)T^{2} \)
67 \( 1 + (0.994 - 0.108i)T^{2} \)
71 \( 1 + (0.735 + 0.0800i)T + (0.976 + 0.214i)T^{2} \)
73 \( 1 + (0.161 + 0.986i)T^{2} \)
79 \( 1 + (-0.634 + 1.59i)T + (-0.725 - 0.687i)T^{2} \)
83 \( 1 + (1.77 - 0.820i)T + (0.647 - 0.762i)T^{2} \)
89 \( 1 + (-0.458 - 0.275i)T + (0.468 + 0.883i)T^{2} \)
97 \( 1 + (0.161 - 0.986i)T^{2} \)
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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

−8.918991070941151777943206166954, −8.483982446181053206223480560757, −7.37281696210791711521122390846, −7.05596828669358032584027451289, −6.16749685747295588909367641996, −5.54907624119518852853414452101, −5.17633413583893671479491401953, −2.43878346979312741807881372122, −1.96053985997086476163484379457, −0.812753519671678995081308545802, 1.28203858751162548036765692147, 2.41422886693926077264926738794, 2.96485240819683765042621613502, 4.26792105794080554256888062662, 5.39223679054217314818851132787, 6.31199067338298073387406511321, 7.00068341504473923653593413486, 8.162873274232874595756642076034, 9.016448786280580226503334479329, 9.435805175933637675872791607558

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