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Properties

Label 2-507-39.8-c1-0-12
Degree $2$
Conductor $507$
Sign $0.614 - 0.789i$
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  + (−0.707 + 0.707i)2-s + (−1 − 1.41i)3-s + 0.999i·4-s + (1.41 − 1.41i)5-s + (1.70 + 0.292i)6-s + (−1 + i)7-s + (−2.12 − 2.12i)8-s + (−1.00 + 2.82i)9-s + 2.00i·10-s + (2.82 + 2.82i)11-s + (1.41 − 0.999i)12-s − 1.41i·14-s + (−3.41 − 0.585i)15-s + 1.00·16-s + (−1.29 − 2.70i)18-s + (−1 − i)19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.577 − 0.816i)3-s + 0.499i·4-s + (0.632 − 0.632i)5-s + (0.696 + 0.119i)6-s + (−0.377 + 0.377i)7-s + (−0.750 − 0.750i)8-s + (−0.333 + 0.942i)9-s + 0.632i·10-s + (0.852 + 0.852i)11-s + (0.408 − 0.288i)12-s − 0.377i·14-s + (−0.881 − 0.151i)15-s + 0.250·16-s + (−0.304 − 0.638i)18-s + (−0.229 − 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.614 - 0.789i$
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: \((2,\ 507,\ (\ :1/2),\ 0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832435 + 0.406990i\)
\(L(\frac12)\) \(\approx\) \(0.832435 + 0.406990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1 + 1.41i)T \)
13 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T - 2iT^{2} \)
5 \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + (-1.41 + 1.41i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (-2.82 - 2.82i)T + 47iT^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (-2.82 - 2.82i)T + 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + (2.82 - 2.82i)T - 71iT^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.65 + 5.65i)T - 83iT^{2} \)
89 \( 1 + (9.89 + 9.89i)T + 89iT^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−11.19990365024464267993970690110, −9.864343013257013424355181694414, −9.056163268291979144109190702399, −8.434294234077526747316891004007, −7.14141003972742684763235584438, −6.75660616510535113707480455451, −5.69794730910731047127424763801, −4.60827072143380032191079711651, −2.86027029231990694911627953365, −1.28217490816909952836313307976, 0.825172695563514558197423061919, 2.66679232346809366379394973186, 3.86625601007857624888927256146, 5.25512262109101644874096447763, 6.16030917518269674007320590691, 6.74557801224362246443472253505, 8.569844693455279759345536947452, 9.362461106491518400430780234348, 10.00695398618436938349196954213, 10.70046423313878735076250485387

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