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Properties

Label 2-507-13.10-c1-0-21
Degree $2$
Conductor $507$
Sign $-0.702 + 0.711i$
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  + (−1.35 + 0.780i)2-s + (−0.5 − 0.866i)3-s + (0.219 − 0.379i)4-s − 3.56i·5-s + (1.35 + 0.780i)6-s + (0.486 + 0.280i)7-s − 2.43i·8-s + (−0.499 + 0.866i)9-s + (2.78 + 4.81i)10-s + (1.73 − i)11-s − 0.438·12-s − 0.876·14-s + (−3.08 + 1.78i)15-s + (2.34 + 4.05i)16-s + (−0.780 + 1.35i)17-s − 1.56i·18-s + ⋯
L(s)  = 1  + (−0.956 + 0.552i)2-s + (−0.288 − 0.499i)3-s + (0.109 − 0.189i)4-s − 1.59i·5-s + (0.552 + 0.318i)6-s + (0.183 + 0.106i)7-s − 0.862i·8-s + (−0.166 + 0.288i)9-s + (0.879 + 1.52i)10-s + (0.522 − 0.301i)11-s − 0.126·12-s − 0.234·14-s + (−0.796 + 0.459i)15-s + (0.585 + 1.01i)16-s + (−0.189 + 0.327i)17-s − 0.368i·18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.702 + 0.711i$
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: \((2,\ 507,\ (\ :1/2),\ -0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166259 - 0.397773i\)
\(L(\frac12)\) \(\approx\) \(0.166259 - 0.397773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 0.780i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + (-0.486 - 0.280i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.780 - 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.16 + 3.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 + 5.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.56iT - 31T^{2} \)
37 \( 1 + (6.54 - 3.78i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.35 - 0.780i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.28 + 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 + (2.49 + 1.43i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.95 + 2.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.1 + 7i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 + (4.22 - 2.43i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.41 - 4.28i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.36400819505776333580002306686, −9.255447071811144264103564732043, −8.661011770969954704861224602356, −8.187491536796611364333417553443, −7.06540268054067223988937769766, −6.15978317691766940834864605552, −5.01329917265759732932361113633, −3.95328059014769722412436031278, −1.68909613561630295014376903440, −0.36701880213898301268550548762, 1.90694529078833871603736934441, 3.13558154817047365711907908753, 4.40726333376805503480558811124, 5.82887978600494103529964122551, 6.75598958321425859481217298175, 7.74989104111435542896793802455, 8.875342156189041009975570503285, 9.665373803782442493535356216590, 10.59911580052115887803855827151, 10.78960381804233993075499579911

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