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Properties

Label 2-507-39.8-c1-0-4
Degree $2$
Conductor $507$
Sign $-0.948 - 0.315i$
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  + (0.707 − 0.707i)2-s + (−1 + 1.41i)3-s + 0.999i·4-s + (−1.41 + 1.41i)5-s + (0.292 + 1.70i)6-s + (−1 + i)7-s + (2.12 + 2.12i)8-s + (−1.00 − 2.82i)9-s + 2.00i·10-s + (−2.82 − 2.82i)11-s + (−1.41 − 0.999i)12-s + 1.41i·14-s + (−0.585 − 3.41i)15-s + 1.00·16-s + (−2.70 − 1.29i)18-s + (−1 − i)19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.577 + 0.816i)3-s + 0.499i·4-s + (−0.632 + 0.632i)5-s + (0.119 + 0.696i)6-s + (−0.377 + 0.377i)7-s + (0.750 + 0.750i)8-s + (−0.333 − 0.942i)9-s + 0.632i·10-s + (−0.852 − 0.852i)11-s + (−0.408 − 0.288i)12-s + 0.377i·14-s + (−0.151 − 0.881i)15-s + 0.250·16-s + (−0.638 − 0.304i)18-s + (−0.229 − 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.948 - 0.315i$
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: \((2,\ 507,\ (\ :1/2),\ -0.948 - 0.315i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108694 + 0.670383i\)
\(L(\frac12)\) \(\approx\) \(0.108694 + 0.670383i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1 - 1.41i)T \)
13 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T - 2iT^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + (2.82 + 2.82i)T + 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + (1.41 - 1.41i)T - 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + (2.82 + 2.82i)T + 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-5 - 5i)T + 67iT^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 + (1 - i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 + (-9.89 - 9.89i)T + 89iT^{2} \)
97 \( 1 + (7 + 7i)T + 97iT^{2} \)
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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

−11.37133306044855106968724093528, −10.67737430389461016997130316578, −9.873635022493393608278915727678, −8.570875686313471260713322572244, −7.83822947520880574577364654393, −6.53910000091238888954378893823, −5.53266345040913408084528041862, −4.42871004452897072754881790645, −3.49778355333802456078703988908, −2.73324232175376469880236674273, 0.35295455859912775921321378120, 1.99684177936964598146125445298, 4.09101466394133302183611782626, 4.94803908289710508373848657373, 5.85971717573553252982609541817, 6.75899751472245250071524594636, 7.59181066812645919498436541365, 8.327790552389090681795235655832, 9.925198743538556442399299197626, 10.42732504324771031940393463089

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