Properties

Label 2-507-13.3-c1-0-18
Degree $2$
Conductor $507$
Sign $0.978 + 0.207i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (0.900 + 1.56i)2-s + (−0.5 − 0.866i)3-s + (−0.623 + 1.07i)4-s − 1.44·5-s + (0.900 − 1.56i)6-s + (1.72 − 2.98i)7-s + 1.35·8-s + (−0.499 + 0.866i)9-s + (−1.30 − 2.25i)10-s + (−2.59 − 4.49i)11-s + 1.24·12-s + 6.20·14-s + (0.722 + 1.25i)15-s + (2.46 + 4.27i)16-s + (0.376 − 0.652i)17-s − 1.80·18-s + ⋯
L(s)  = 1  + (0.637 + 1.10i)2-s + (−0.288 − 0.499i)3-s + (−0.311 + 0.539i)4-s − 0.646·5-s + (0.367 − 0.637i)6-s + (0.651 − 1.12i)7-s + 0.479·8-s + (−0.166 + 0.288i)9-s + (−0.411 − 0.713i)10-s + (−0.781 − 1.35i)11-s + 0.359·12-s + 1.65·14-s + (0.186 + 0.323i)15-s + (0.617 + 1.06i)16-s + (0.0913 − 0.158i)17-s − 0.424·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70936 - 0.179192i\)
\(L(\frac12)\) \(\approx\) \(1.70936 - 0.179192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.900 - 1.56i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + (-1.72 + 2.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.376 + 0.652i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.98 + 6.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.41 - 2.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.95 - 3.38i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + (-3.12 - 5.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.900 - 1.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 + (-0.939 + 1.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.67 - 2.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.27 + 3.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.55 - 7.89i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.95T + 73T^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 - 6.46T + 83T^{2} \)
89 \( 1 + (-0.579 - 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.32 + 7.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.17440214517562505000376443311, −10.21284128120590795170921914349, −8.570436559218303962183993248812, −7.73611830280630037618510083297, −7.30291182013175935761545968360, −6.33164857214417713888953721947, −5.25277623105440961769563329001, −4.55679021945064720091158865421, −3.21826261881645412198445317402, −0.941650694877956833792088186429, 1.86656844632471711841863355583, 2.97916692576758770219796862989, 4.22023033205328564997282158615, 4.89654716539588817058792672746, 5.86001939548272916020245456512, 7.57196456542622143676985425597, 8.170086836748913390133312944963, 9.617888985695651301964036845969, 10.22252117783289147623720393896, 11.18259502096731242462983948479

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