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Properties

Label 2-507-507.314-c1-0-41
Degree $2$
Conductor $507$
Sign $0.986 + 0.161i$
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  + (1.70 − 0.277i)3-s + (−0.783 + 1.83i)4-s + (1.45 − 4.07i)7-s + (2.84 − 0.950i)9-s + (−0.829 + 3.36i)12-s + (2.59 − 2.5i)13-s + (−2.77 − 2.88i)16-s + (2.01 + 7.51i)19-s + (1.35 − 7.37i)21-s + (2.32 − 4.42i)25-s + (4.60 − 2.41i)27-s + (6.36 + 5.86i)28-s + (−3.22 + 10.3i)31-s + (−0.482 + 5.98i)36-s + (1.88 + 8.37i)37-s + ⋯
L(s)  = 1  + (0.987 − 0.160i)3-s + (−0.391 + 0.919i)4-s + (0.549 − 1.54i)7-s + (0.948 − 0.316i)9-s + (−0.239 + 0.970i)12-s + (0.720 − 0.693i)13-s + (−0.692 − 0.721i)16-s + (0.461 + 1.72i)19-s + (0.294 − 1.60i)21-s + (0.464 − 0.885i)25-s + (0.885 − 0.464i)27-s + (1.20 + 1.10i)28-s + (−0.579 + 1.86i)31-s + (−0.0804 + 0.996i)36-s + (0.310 + 1.37i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.986 + 0.161i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97515 - 0.160735i\)
\(L(\frac12)\) \(\approx\) \(1.97515 - 0.160735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.277i)T \)
13 \( 1 + (-2.59 + 2.5i)T \)
good2 \( 1 + (0.783 - 1.83i)T^{2} \)
5 \( 1 + (-2.32 + 4.42i)T^{2} \)
7 \( 1 + (-1.45 + 4.07i)T + (-5.42 - 4.42i)T^{2} \)
11 \( 1 + (6.60 - 8.79i)T^{2} \)
17 \( 1 + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (-2.01 - 7.51i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (26.6 + 11.3i)T^{2} \)
31 \( 1 + (3.22 - 10.3i)T + (-25.5 - 17.6i)T^{2} \)
37 \( 1 + (-1.88 - 8.37i)T + (-33.4 + 15.8i)T^{2} \)
41 \( 1 + (12.9 + 38.8i)T^{2} \)
43 \( 1 + (6.24 + 9.87i)T + (-18.4 + 38.8i)T^{2} \)
47 \( 1 + (-11.2 + 45.6i)T^{2} \)
53 \( 1 + (18.7 + 49.5i)T^{2} \)
59 \( 1 + (-58.9 + 2.37i)T^{2} \)
61 \( 1 + (13.9 - 1.12i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (-5.26 + 13.0i)T + (-48.3 - 46.4i)T^{2} \)
71 \( 1 + (-69.5 - 14.2i)T^{2} \)
73 \( 1 + (13.6 + 0.824i)T + (72.4 + 8.79i)T^{2} \)
79 \( 1 + (0.946 - 7.79i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (-55.0 - 62.1i)T^{2} \)
89 \( 1 + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.79 - 8.70i)T + (-51.8 + 81.9i)T^{2} \)
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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

−10.60296721121306359614322068935, −10.10206079756179844712176105425, −8.809839096397488251668748349324, −8.105883550016749507773802177431, −7.59676946546447508235936801526, −6.65725307178406723051405991569, −4.85515725906551587674849723950, −3.82005583383568157989123675984, −3.24588477016990885633344233603, −1.37164089712856590189836885180, 1.66817877605483211602054274766, 2.74208141395669801631575946854, 4.30583862292039927312080189366, 5.20911162378944297421412948247, 6.17945709413530130904142422228, 7.44096541707080494951544503072, 8.670578353013673571593303509579, 9.101274372611613947856918706292, 9.651140867297213662734054656681, 11.03629326309815383751290411544

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