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Properties

Label 2-507-39.5-c1-0-14
Degree $2$
Conductor $507$
Sign $-0.0125 - 0.999i$
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 + (−2.25 − 0.293i)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 + (−5.16 − 0.671i)15-s + 3.37·16-s + 0.198·17-s + (3.40 − 1.98i)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 + (−0.920 − 0.119i)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 + (−1.33 − 0.173i)15-s + 0.842·16-s + 0.0482·17-s + (0.802 − 0.466i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.0125 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.0125 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43186 + 1.44994i\)
\(L(\frac12)\) \(\approx\) \(1.43186 + 1.44994i\)
\(L(\frac{3}{2})\) 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

−11.00036140137645342390140100966, −10.34662819234942939477266290546, −9.619545733537231157799732012086, −8.470319233434678909781224306203, −6.98088903781774988601365790201, −6.13399640481942184408801124178, −5.76202590096403062691811913157, −4.80536607738604064787716223104, −3.62109483445524268991347905491, −1.81407869116434658819789541082, 1.32767855862334024971008426756, 2.20940547336661745047875790972, 4.25730382377750002943264211063, 4.78052172749322086259042023364, 5.77319312371933257610586110926, 6.96220784112052689657845979392, 7.902231804409031009363809678894, 8.885690847205571661293027743411, 10.18018039723633150654700129585, 10.88261854811920375819331984979

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