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Properties

Label 2-507-39.5-c1-0-32
Degree $2$
Conductor $507$
Sign $0.957 + 0.289i$
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  + (1.06 + 1.06i)2-s + 1.73·3-s + 0.267i·4-s + (−2.90 − 2.90i)5-s + (1.84 + 1.84i)6-s + (1.84 − 1.84i)8-s + 2.99·9-s − 6.19i·10-s + (0.779 − 0.779i)11-s + 0.464i·12-s + (−5.03 − 5.03i)15-s + 4.46·16-s + (3.19 + 3.19i)18-s + (0.779 − 0.779i)20-s + 1.66·22-s + ⋯
L(s)  = 1  + (0.752 + 0.752i)2-s + 1.00·3-s + 0.133i·4-s + (−1.30 − 1.30i)5-s + (0.752 + 0.752i)6-s + (0.652 − 0.652i)8-s + 0.999·9-s − 1.95i·10-s + (0.235 − 0.235i)11-s + 0.133i·12-s + (−1.30 − 1.30i)15-s + 1.11·16-s + (0.752 + 0.752i)18-s + (0.174 − 0.174i)20-s + 0.353·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45451 - 0.363436i\)
\(L(\frac12)\) \(\approx\) \(2.45451 - 0.363436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
13 \( 1 \)
good2 \( 1 + (-1.06 - 1.06i)T + 2iT^{2} \)
5 \( 1 + (2.90 + 2.90i)T + 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + (-0.779 + 0.779i)T - 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 + (-7.16 - 7.16i)T + 41iT^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (6.59 - 6.59i)T - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (10.8 - 10.8i)T - 59iT^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + (-3.47 - 3.47i)T + 71iT^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-9.29 - 9.29i)T + 83iT^{2} \)
89 \( 1 + (-12.9 + 12.9i)T - 89iT^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.96153468768485494933108516117, −9.660891145136794710849712065911, −8.884906043216554664008814813249, −7.953334757938382006662711115680, −7.47341803071928908747098009077, −6.23679127626334903266748623938, −4.87280803711042296969604880660, −4.32898921763266576370035679932, −3.37822004936429757339339350209, −1.22356552749257777189594829339, 2.19175530138706615587672789831, 3.21047575916616932078415665418, 3.77806944038316498668913512194, 4.67847535034410811164496398790, 6.57065309956597167510790212615, 7.57199594608092979575865022343, 7.973193756415000563126603972191, 9.197525970115304506128053775749, 10.49302140225007285781647899654, 10.94753954845737971996985281060

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