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Properties

Label 2-507-39.8-c1-0-25
Degree $2$
Conductor $507$
Sign $0.774 + 0.633i$
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  + (1.42 − 1.42i)2-s + (−1.57 + 0.730i)3-s − 2.07i·4-s + (−1.72 + 1.72i)5-s + (−1.19 + 3.28i)6-s + (2.20 − 2.20i)7-s + (−0.105 − 0.105i)8-s + (1.93 − 2.29i)9-s + 4.91i·10-s + (1.95 + 1.95i)11-s + (1.51 + 3.25i)12-s − 6.28i·14-s + (1.44 − 3.96i)15-s + 3.84·16-s + 5.78·17-s + (−0.515 − 6.03i)18-s + ⋯
L(s)  = 1  + (1.00 − 1.00i)2-s + (−0.906 + 0.421i)3-s − 1.03i·4-s + (−0.770 + 0.770i)5-s + (−0.489 + 1.34i)6-s + (0.831 − 0.831i)7-s + (−0.0372 − 0.0372i)8-s + (0.644 − 0.764i)9-s + 1.55i·10-s + (0.590 + 0.590i)11-s + (0.437 + 0.940i)12-s − 1.67i·14-s + (0.373 − 1.02i)15-s + 0.961·16-s + 1.40·17-s + (−0.121 − 1.42i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.774 + 0.633i$
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: \((2,\ 507,\ (\ :1/2),\ 0.774 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79186 - 0.639423i\)
\(L(\frac12)\) \(\approx\) \(1.79186 - 0.639423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 - 0.730i)T \)
13 \( 1 \)
good2 \( 1 + (-1.42 + 1.42i)T - 2iT^{2} \)
5 \( 1 + (1.72 - 1.72i)T - 5iT^{2} \)
7 \( 1 + (-2.20 + 2.20i)T - 7iT^{2} \)
11 \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \)
17 \( 1 - 5.78T + 17T^{2} \)
19 \( 1 + (1.06 + 1.06i)T + 19iT^{2} \)
23 \( 1 - 3.86T + 23T^{2} \)
29 \( 1 + 2.92iT - 29T^{2} \)
31 \( 1 + (3.56 + 3.56i)T + 31iT^{2} \)
37 \( 1 + (-2.83 + 2.83i)T - 37iT^{2} \)
41 \( 1 + (4.79 - 4.79i)T - 41iT^{2} \)
43 \( 1 - 1.84iT - 43T^{2} \)
47 \( 1 + (0.115 + 0.115i)T + 47iT^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + (-1.22 - 1.22i)T + 59iT^{2} \)
61 \( 1 + 5.39T + 61T^{2} \)
67 \( 1 + (9.65 + 9.65i)T + 67iT^{2} \)
71 \( 1 + (-0.239 + 0.239i)T - 71iT^{2} \)
73 \( 1 + (8.54 - 8.54i)T - 73iT^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (2.83 - 2.83i)T - 83iT^{2} \)
89 \( 1 + (3.00 + 3.00i)T + 89iT^{2} \)
97 \( 1 + (6.99 + 6.99i)T + 97iT^{2} \)
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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

−11.02142288719728836969598785644, −10.50374052867413355052432587335, −9.561501769877840750299592818149, −7.79543733534912276551449659566, −7.13369435814905592727878262304, −5.81827635800593443335792933554, −4.69110508315771051708098102059, −4.10324118765897944522979390710, −3.19353579568444494732536899211, −1.33931044664338234980189578191, 1.27829845630712448297948714021, 3.64061873965735754663379113210, 4.83754974657041812665499884043, 5.31286516382125242849139644259, 6.15446743376652949217618864854, 7.20750121353939892399681900905, 8.021485744495099508796468501564, 8.781952332022971371695526680883, 10.34174902501426542292018411648, 11.46775778735075068021606771052

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