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Properties

Label 2-513-19.7-c1-0-4
Degree $2$
Conductor $513$
Sign $-0.999 - 0.0424i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 + 2.16i)2-s + (−2.12 + 3.67i)4-s + (0.535 + 0.927i)5-s − 0.193·7-s − 5.61·8-s + (−1.33 + 2.31i)10-s + 0.571·11-s + (−2.84 + 4.92i)13-s + (−0.241 − 0.418i)14-s + (−2.77 − 4.80i)16-s + (−1.02 − 1.77i)17-s + (−1.50 + 4.08i)19-s − 4.54·20-s + (0.714 + 1.23i)22-s + (0.777 − 1.34i)23-s + ⋯
L(s)  = 1  + (0.883 + 1.53i)2-s + (−1.06 + 1.83i)4-s + (0.239 + 0.414i)5-s − 0.0731·7-s − 1.98·8-s + (−0.423 + 0.732i)10-s + 0.172·11-s + (−0.788 + 1.36i)13-s + (−0.0646 − 0.111i)14-s + (−0.693 − 1.20i)16-s + (−0.248 − 0.429i)17-s + (−0.345 + 0.938i)19-s − 1.01·20-s + (0.152 + 0.263i)22-s + (0.162 − 0.280i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.999 - 0.0424i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.999 - 0.0424i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0428841 + 2.01905i\)
\(L(\frac12)\) \(\approx\) \(0.0428841 + 2.01905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (1.50 - 4.08i)T \)
good2 \( 1 + (-1.24 - 2.16i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.535 - 0.927i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.193T + 7T^{2} \)
11 \( 1 - 0.571T + 11T^{2} \)
13 \( 1 + (2.84 - 4.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.02 + 1.77i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.777 + 1.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.04 + 5.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.86T + 31T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 + (0.964 + 1.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.60 - 2.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.30 + 9.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.84 - 6.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.89 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.01 - 6.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 5.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.50 + 6.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.03 - 6.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.44 - 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.02T + 83T^{2} \)
89 \( 1 + (-8.79 + 15.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.79 - 6.57i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.70048052945830478602780686670, −10.29867609919975925033991877751, −9.311433019317717362486332504900, −8.330060206099331312719105714846, −7.42147756216881920533388794998, −6.56772246894755268757802769447, −6.06441272897424677841236923784, −4.73996746447490160029416058608, −4.12470408436967608020279248208, −2.57677129702440396671365549528, 0.956250168749898855938373201905, 2.44925504021124640207161932027, 3.34928075268671497688962197201, 4.67165790119698596606100179957, 5.22413903880472668562739293471, 6.42083225224702747571683028608, 7.909365132884300286144009023113, 9.126457671415484765968836958831, 9.845256493615956485906097629572, 10.69245742391578599086139721845

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