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Properties

Label 2-507-13.4-c1-0-24
Degree $2$
Conductor $507$
Sign $-0.711 + 0.702i$
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  + (0.5 − 0.866i)3-s + (−1 − 1.73i)4-s − 3.46i·5-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (3 + 1.73i)11-s − 1.99·12-s + (−2.99 − 1.73i)15-s + (−1.99 + 3.46i)16-s + (−3 + 1.73i)19-s + (−5.99 + 3.46i)20-s − 1.73i·21-s + (3 − 5.19i)23-s − 6.99·25-s − 0.999·27-s + (−3 − 1.73i)28-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.5 − 0.866i)4-s − 1.54i·5-s + (0.566 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.904 + 0.522i)11-s − 0.577·12-s + (−0.774 − 0.447i)15-s + (−0.499 + 0.866i)16-s + (−0.688 + 0.397i)19-s + (−1.34 + 0.774i)20-s − 0.377i·21-s + (0.625 − 1.08i)23-s − 1.39·25-s − 0.192·27-s + (−0.566 − 0.327i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.542741 - 1.32226i\)
\(L(\frac12)\) \(\approx\) \(0.542741 - 1.32226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-3 + 1.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 + 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9 + 5.19i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + (6 + 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.49442440648619350114865378177, −9.406203806181377274711800818783, −8.862486955937312646303034722741, −8.144735834633521420420117327801, −6.89307824585006270304523573160, −5.78878743226053558966862848467, −4.76750174965205353128656366852, −4.13859799845170209925812116737, −1.82616157343769518731569615942, −0.906293462322824100179264880209, 2.45271418908805836296304860368, 3.44685754561145071373143351144, 4.23481505860246828027979729659, 5.65401343942211927856765120769, 6.86125411285924226070362454242, 7.64979674196873930357252209275, 8.656125750921008090335996092206, 9.347911968868255416253747207021, 10.41221220614766045981586302825, 11.36147182238873260024867362005

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