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Properties

Label 2-507-13.10-c1-0-9
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: $\chi_{507} (361, \cdot )$
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

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

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