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Properties

Label 2-507-39.8-c1-0-20
Degree $2$
Conductor $507$
Sign $0.965 - 0.259i$
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  + (0.928 − 0.928i)2-s + (−1.37 + 1.05i)3-s + 0.276i·4-s + (2.12 − 2.12i)5-s + (−0.293 + 2.25i)6-s + (−2.06 + 2.06i)7-s + (2.11 + 2.11i)8-s + (0.767 − 2.90i)9-s − 3.94i·10-s + (1.88 + 1.88i)11-s + (−0.291 − 0.379i)12-s + 3.83i·14-s + (−0.671 + 5.16i)15-s + 3.37·16-s − 0.198·17-s + (−1.98 − 3.40i)18-s + ⋯
L(s)  = 1  + (0.656 − 0.656i)2-s + (−0.792 + 0.610i)3-s + 0.138i·4-s + (0.950 − 0.950i)5-s + (−0.119 + 0.920i)6-s + (−0.780 + 0.780i)7-s + (0.747 + 0.747i)8-s + (0.255 − 0.966i)9-s − 1.24i·10-s + (0.568 + 0.568i)11-s + (−0.0842 − 0.109i)12-s + 1.02i·14-s + (−0.173 + 1.33i)15-s + 0.842·16-s − 0.0482·17-s + (−0.466 − 0.802i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.965 - 0.259i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79193 + 0.236084i\)
\(L(\frac12)\) \(\approx\) \(1.79193 + 0.236084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.37 - 1.05i)T \)
13 \( 1 \)
good2 \( 1 + (-0.928 + 0.928i)T - 2iT^{2} \)
5 \( 1 + (-2.12 + 2.12i)T - 5iT^{2} \)
7 \( 1 + (2.06 - 2.06i)T - 7iT^{2} \)
11 \( 1 + (-1.88 - 1.88i)T + 11iT^{2} \)
17 \( 1 + 0.198T + 17T^{2} \)
19 \( 1 + (-3.75 - 3.75i)T + 19iT^{2} \)
23 \( 1 - 3.30T + 23T^{2} \)
29 \( 1 - 3.73iT - 29T^{2} \)
31 \( 1 + (0.550 + 0.550i)T + 31iT^{2} \)
37 \( 1 + (-3.60 + 3.60i)T - 37iT^{2} \)
41 \( 1 + (2.69 - 2.69i)T - 41iT^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 + (-4.29 - 4.29i)T + 47iT^{2} \)
53 \( 1 + 13.6iT - 53T^{2} \)
59 \( 1 + (2.36 + 2.36i)T + 59iT^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 + (5.33 + 5.33i)T + 67iT^{2} \)
71 \( 1 + (3.78 - 3.78i)T - 71iT^{2} \)
73 \( 1 + (3.46 - 3.46i)T - 73iT^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + (1.84 - 1.84i)T - 83iT^{2} \)
89 \( 1 + (0.776 + 0.776i)T + 89iT^{2} \)
97 \( 1 + (8.60 + 8.60i)T + 97iT^{2} \)
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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

−11.13548340639462754616006905889, −10.03681053908892688070753419733, −9.421419992869075759833900968232, −8.666089981826169615316495249256, −7.05358218465488326408662529204, −5.80284436050016228164960495061, −5.27492414129221993127854834204, −4.27320123517215027081174202752, −3.15179902007895212774299644546, −1.62849816984869880130125900883, 1.12555489739670202746274407457, 2.93592733949866323743869805305, 4.43429388668340967861211594193, 5.63581534618560025286642364462, 6.29103767647957363765158690278, 6.84536238698394073635330151744, 7.52144275647243516642208017625, 9.344516899359635426674639251194, 10.19929229634608327153552639053, 10.79090756131629348014210424809

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