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Properties

Label 2-507-39.8-c1-0-18
Degree $2$
Conductor $507$
Sign $0.497 + 0.867i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.497 + 0.867i$
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: \((2,\ 507,\ (\ :1/2),\ 0.497 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0917338 - 0.0531451i\)
\(L(\frac12)\) \(\approx\) \(0.0917338 - 0.0531451i\)
\(L(\frac{3}{2})\) 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

−10.76992780659388211863507706269, −9.884598558101802661725099028262, −8.707432561831062956090640242000, −7.70494437054298806629657205951, −7.36037721458523387136767230407, −6.25707301724719839826739336414, −5.70519027638343115283148177350, −3.79548070230829444968208514754, −2.72807416867231561125100408115, −0.10896891289871984444092751454, 1.01371705377925951336707659220, 3.19137322810052901662419461409, 4.43247586753963745142385217509, 5.18147337270740276262574865689, 6.42645180467820329527474645570, 7.59290847080471802210849811653, 8.668538853858444152694965232639, 9.726199219080748102053465016715, 9.971798182028565513374012035033, 11.10543059844855350643669347598

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