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Properties

Label 2-507-507.92-c0-0-0
Degree $2$
Conductor $507$
Sign $0.956 - 0.293i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.970 + 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.234 + 1.92i)7-s + (0.885 − 0.464i)9-s + (−0.354 + 0.935i)12-s + 13-s + (−0.354 − 0.935i)16-s + 1.77·19-s + (−0.234 − 1.92i)21-s + (−0.748 − 0.663i)25-s + (−0.748 + 0.663i)27-s + (1.45 + 1.28i)28-s + (−0.850 + 0.753i)31-s + (0.120 − 0.992i)36-s + (−0.180 + 0.159i)37-s + ⋯
L(s)  = 1  + (−0.970 + 0.239i)3-s + (0.568 − 0.822i)4-s + (−0.234 + 1.92i)7-s + (0.885 − 0.464i)9-s + (−0.354 + 0.935i)12-s + 13-s + (−0.354 − 0.935i)16-s + 1.77·19-s + (−0.234 − 1.92i)21-s + (−0.748 − 0.663i)25-s + (−0.748 + 0.663i)27-s + (1.45 + 1.28i)28-s + (−0.850 + 0.753i)31-s + (0.120 − 0.992i)36-s + (−0.180 + 0.159i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.956 - 0.293i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.956 - 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7686494847\)
\(L(\frac12)\) \(\approx\) \(0.7686494847\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.568 + 0.822i)T^{2} \)
5 \( 1 + (0.748 + 0.663i)T^{2} \)
7 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
11 \( 1 + (-0.568 - 0.822i)T^{2} \)
17 \( 1 + (0.970 + 0.239i)T^{2} \)
19 \( 1 - 1.77T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.568 + 0.822i)T^{2} \)
31 \( 1 + (0.850 - 0.753i)T + (0.120 - 0.992i)T^{2} \)
37 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
41 \( 1 + (-0.885 + 0.464i)T^{2} \)
43 \( 1 + (0.180 + 0.159i)T + (0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.354 - 0.935i)T^{2} \)
53 \( 1 + (0.970 + 0.239i)T^{2} \)
59 \( 1 + (0.748 + 0.663i)T^{2} \)
61 \( 1 + (-0.136 - 1.12i)T + (-0.970 + 0.239i)T^{2} \)
67 \( 1 + (0.850 + 1.23i)T + (-0.354 + 0.935i)T^{2} \)
71 \( 1 + (-0.885 + 0.464i)T^{2} \)
73 \( 1 + (0.627 + 0.329i)T + (0.568 + 0.822i)T^{2} \)
79 \( 1 + (1.10 + 1.59i)T + (-0.354 + 0.935i)T^{2} \)
83 \( 1 + (-0.885 - 0.464i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.709 + 1.87i)T + (-0.748 + 0.663i)T^{2} \)
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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

−11.37480689164685041858002155045, −10.32280113210484243380263115228, −9.536716622390145153281098338257, −8.766511144030860362924136227186, −7.29335542665629170709419710505, −6.11826088067169385274756357918, −5.77667045444731869965367310282, −4.95875058919796436949082368419, −3.17424122998794333708033990559, −1.67053522273250602204770825254, 1.34442042269453078343376898894, 3.45822979233944013630701931225, 4.18207969391644284284800194303, 5.62364926347831532353318854844, 6.73082953663368930372608113173, 7.34798083066179391427067417751, 7.941519140183777879144216333869, 9.555396354633776199697322843177, 10.46821351249743534831472385651, 11.22906133625679474737579742236

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