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Properties

Label 2-507-39.8-c1-0-7
Degree $2$
Conductor $507$
Sign $0.793 - 0.609i$
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  + (1.42 − 1.42i)2-s + (−1.57 − 0.730i)3-s − 2.07i·4-s + (−1.72 + 1.72i)5-s + (−3.28 + 1.19i)6-s + (−2.20 + 2.20i)7-s + (−0.105 − 0.105i)8-s + (1.93 + 2.29i)9-s + 4.91i·10-s + (1.95 + 1.95i)11-s + (−1.51 + 3.25i)12-s + 6.28i·14-s + (3.96 − 1.44i)15-s + 3.84·16-s − 5.78·17-s + (6.03 + 0.515i)18-s + ⋯
L(s)  = 1  + (1.00 − 1.00i)2-s + (−0.906 − 0.421i)3-s − 1.03i·4-s + (−0.770 + 0.770i)5-s + (−1.34 + 0.489i)6-s + (−0.831 + 0.831i)7-s + (−0.0372 − 0.0372i)8-s + (0.644 + 0.764i)9-s + 1.55i·10-s + (0.590 + 0.590i)11-s + (−0.437 + 0.940i)12-s + 1.67i·14-s + (1.02 − 0.373i)15-s + 0.961·16-s − 1.40·17-s + (1.42 + 0.121i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.793 - 0.609i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.793 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04695 + 0.355707i\)
\(L(\frac12)\) \(\approx\) \(1.04695 + 0.355707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 + 0.730i)T \)
13 \( 1 \)
good2 \( 1 + (-1.42 + 1.42i)T - 2iT^{2} \)
5 \( 1 + (1.72 - 1.72i)T - 5iT^{2} \)
7 \( 1 + (2.20 - 2.20i)T - 7iT^{2} \)
11 \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \)
17 \( 1 + 5.78T + 17T^{2} \)
19 \( 1 + (-1.06 - 1.06i)T + 19iT^{2} \)
23 \( 1 + 3.86T + 23T^{2} \)
29 \( 1 - 2.92iT - 29T^{2} \)
31 \( 1 + (-3.56 - 3.56i)T + 31iT^{2} \)
37 \( 1 + (2.83 - 2.83i)T - 37iT^{2} \)
41 \( 1 + (4.79 - 4.79i)T - 41iT^{2} \)
43 \( 1 - 1.84iT - 43T^{2} \)
47 \( 1 + (0.115 + 0.115i)T + 47iT^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 + (-1.22 - 1.22i)T + 59iT^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 + (-9.65 - 9.65i)T + 67iT^{2} \)
71 \( 1 + (-0.239 + 0.239i)T - 71iT^{2} \)
73 \( 1 + (-8.54 + 8.54i)T - 73iT^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (2.83 - 2.83i)T - 83iT^{2} \)
89 \( 1 + (3.00 + 3.00i)T + 89iT^{2} \)
97 \( 1 + (-6.99 - 6.99i)T + 97iT^{2} \)
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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

−11.36348729537492801200191523472, −10.55692608861775434596724046648, −9.669661365212134044998965726074, −8.221432896018161390705136078425, −6.94471850281235635172584539669, −6.34800644953153217712093862938, −5.15842802574907486343980729454, −4.16900083788591100687938837943, −3.10946085104555243997490083097, −1.92732505154574921012450754784, 0.53141490687424927440104581547, 3.78409048546545702685968565633, 4.17030359672487219025274505176, 5.11375578730681417932787679502, 6.24620031313493185677524093137, 6.72942574095093193738735400253, 7.76640544330348340622272147988, 8.953736091615938913611032904255, 10.00575752567544440678912956825, 10.96940308699244473790698377263

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