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Properties

Label 2-507-39.8-c1-0-15
Degree $2$
Conductor $507$
Sign $0.579 - 0.814i$
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  + (1.38 − 1.38i)2-s + (0.526 + 1.65i)3-s − 1.83i·4-s + (−1.04 + 1.04i)5-s + (3.01 + 1.55i)6-s + (−3.17 + 3.17i)7-s + (0.233 + 0.233i)8-s + (−2.44 + 1.73i)9-s + 2.89i·10-s + (−0.108 − 0.108i)11-s + (3.02 − 0.964i)12-s + 8.77i·14-s + (−2.27 − 1.17i)15-s + 4.30·16-s + 3.16·17-s + (−0.980 + 5.78i)18-s + ⋯
L(s)  = 1  + (0.978 − 0.978i)2-s + (0.303 + 0.952i)3-s − 0.915i·4-s + (−0.468 + 0.468i)5-s + (1.22 + 0.634i)6-s + (−1.19 + 1.19i)7-s + (0.0825 + 0.0825i)8-s + (−0.815 + 0.579i)9-s + 0.916i·10-s + (−0.0326 − 0.0326i)11-s + (0.872 − 0.278i)12-s + 2.34i·14-s + (−0.588 − 0.303i)15-s + 1.07·16-s + 0.768·17-s + (−0.231 + 1.36i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.579 - 0.814i$
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: \((2,\ 507,\ (\ :1/2),\ 0.579 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81332 + 0.935258i\)
\(L(\frac12)\) \(\approx\) \(1.81332 + 0.935258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.526 - 1.65i)T \)
13 \( 1 \)
good2 \( 1 + (-1.38 + 1.38i)T - 2iT^{2} \)
5 \( 1 + (1.04 - 1.04i)T - 5iT^{2} \)
7 \( 1 + (3.17 - 3.17i)T - 7iT^{2} \)
11 \( 1 + (0.108 + 0.108i)T + 11iT^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 + (0.846 + 0.846i)T + 19iT^{2} \)
23 \( 1 - 6.70T + 23T^{2} \)
29 \( 1 + 1.98iT - 29T^{2} \)
31 \( 1 + (3.64 + 3.64i)T + 31iT^{2} \)
37 \( 1 + (2.31 - 2.31i)T - 37iT^{2} \)
41 \( 1 + (-5.91 + 5.91i)T - 41iT^{2} \)
43 \( 1 - 2.78iT - 43T^{2} \)
47 \( 1 + (-4.06 - 4.06i)T + 47iT^{2} \)
53 \( 1 + 0.628iT - 53T^{2} \)
59 \( 1 + (-6.20 - 6.20i)T + 59iT^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 + (0.475 + 0.475i)T + 67iT^{2} \)
71 \( 1 + (8.09 - 8.09i)T - 71iT^{2} \)
73 \( 1 + (-3.92 + 3.92i)T - 73iT^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 + (-6.40 + 6.40i)T - 83iT^{2} \)
89 \( 1 + (3.75 + 3.75i)T + 89iT^{2} \)
97 \( 1 + (3.16 + 3.16i)T + 97iT^{2} \)
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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

−11.18978586178955641802597862845, −10.35741725401942057471969256862, −9.489149506664044349676974911065, −8.717333995303473818356717220101, −7.42309636216186714630590746615, −5.92661751405332503747489858758, −5.19739626145848607074559782788, −3.95052915189438096998850840148, −3.16509665583968610842382288514, −2.53361712702852749215074473326, 0.895400335625232184103665924342, 3.20721975887126819629629739977, 4.00705122145223502473582051371, 5.27253533259263845119269220241, 6.34813756809823967907636091865, 7.05536288713053075674596237976, 7.58222367507323811909426208135, 8.638687364451293047153691616455, 9.807463562898345518153356881858, 10.85469117605361213724625160341

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