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Properties

Label 2-507-507.206-c1-0-34
Degree $2$
Conductor $507$
Sign $0.974 - 0.222i$
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  + (1.72 + 0.139i)3-s + (1.95 − 0.400i)4-s + (−0.421 + 1.87i)7-s + (2.96 + 0.481i)9-s + (3.43 − 0.417i)12-s + (−2.59 + 2.5i)13-s + (3.67 − 1.56i)16-s + (0.120 − 0.0321i)19-s + (−0.989 + 3.17i)21-s + (−1.19 − 4.85i)25-s + (5.04 + 1.24i)27-s + (−0.0773 + 3.83i)28-s + (−3.88 − 6.42i)31-s + (5.99 − 0.241i)36-s + (−6.55 − 3.61i)37-s + ⋯
L(s)  = 1  + (0.996 + 0.0804i)3-s + (0.979 − 0.200i)4-s + (−0.159 + 0.708i)7-s + (0.987 + 0.160i)9-s + (0.992 − 0.120i)12-s + (−0.720 + 0.693i)13-s + (0.919 − 0.391i)16-s + (0.0275 − 0.00738i)19-s + (−0.215 + 0.692i)21-s + (−0.239 − 0.970i)25-s + (0.970 + 0.239i)27-s + (−0.0146 + 0.725i)28-s + (−0.697 − 1.15i)31-s + (0.999 − 0.0402i)36-s + (−1.07 − 0.593i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (206, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36047 + 0.266533i\)
\(L(\frac12)\) \(\approx\) \(2.36047 + 0.266533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.139i)T \)
13 \( 1 + (2.59 - 2.5i)T \)
good2 \( 1 + (-1.95 + 0.400i)T^{2} \)
5 \( 1 + (1.19 + 4.85i)T^{2} \)
7 \( 1 + (0.421 - 1.87i)T + (-6.32 - 3.00i)T^{2} \)
11 \( 1 + (3.48 + 10.4i)T^{2} \)
17 \( 1 + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (-0.120 + 0.0321i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.80 + 28.4i)T^{2} \)
31 \( 1 + (3.88 + 6.42i)T + (-14.4 + 27.4i)T^{2} \)
37 \( 1 + (6.55 + 3.61i)T + (19.7 + 31.2i)T^{2} \)
41 \( 1 + (6.57 - 40.4i)T^{2} \)
43 \( 1 + (4.32 + 1.25i)T + (36.3 + 22.9i)T^{2} \)
47 \( 1 + (-46.6 + 5.66i)T^{2} \)
53 \( 1 + (-30.1 - 43.6i)T^{2} \)
59 \( 1 + (-42.5 + 40.8i)T^{2} \)
61 \( 1 + (-0.0861 + 2.13i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (2.62 - 1.73i)T + (26.2 - 61.6i)T^{2} \)
71 \( 1 + (54.9 + 44.9i)T^{2} \)
73 \( 1 + (-3.74 - 8.31i)T + (-48.4 + 54.6i)T^{2} \)
79 \( 1 + (-2.85 - 2.52i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (77.6 + 29.4i)T^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.38 - 16.8i)T + (-93.1 + 26.9i)T^{2} \)
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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

−10.86543213088147423132169760526, −9.928560415454794590959168334404, −9.235199295684174002524964946514, −8.235217384402544707452620578739, −7.33258226824249619369535034077, −6.53088917230831205791993074522, −5.36133295749842327813601350586, −3.97785018430427767902431715178, −2.69081305680515805610628920475, −1.94172842144460460374786611610, 1.61851043477531308351887521478, 2.92757571466167796821189147929, 3.71162628697647093087638564666, 5.18949947075380622956232050030, 6.65790712564906372176759190111, 7.32844998537193144410703901922, 7.984318676544506265235693506900, 9.055615422211811205553845792901, 10.14193045896289850109962824662, 10.63931007119483716558270119528

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