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Properties

Label 2-507-13.12-c1-0-18
Degree $2$
Conductor $507$
Sign $0.277 + 0.960i$
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  − 3-s + 2·4-s − 3.46i·5-s + 1.73i·7-s + 9-s − 3.46i·11-s − 2·12-s + 3.46i·15-s + 4·16-s − 3.46i·19-s − 6.92i·20-s − 1.73i·21-s − 6·23-s − 6.99·25-s − 27-s + 3.46i·28-s + ⋯
L(s)  = 1  − 0.577·3-s + 4-s − 1.54i·5-s + 0.654i·7-s + 0.333·9-s − 1.04i·11-s − 0.577·12-s + 0.894i·15-s + 16-s − 0.794i·19-s − 1.54i·20-s − 0.377i·21-s − 1.25·23-s − 1.39·25-s − 0.192·27-s + 0.654i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14227 - 0.859169i\)
\(L(\frac12)\) \(\approx\) \(1.14227 - 0.859169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 2T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 - 3.46iT - 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 8.66iT - 67T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−10.88459825257715122944784025758, −9.912383991182912542351654698313, −8.768005058525067394405677337667, −8.229261947696721318114790512853, −6.95468585242479492883352039003, −5.83579902619251598800908848119, −5.37784818348028851549801574913, −4.05798604057711294781276368355, −2.41577920338475777288576920701, −0.936485251864389765306096879002, 1.86463488811903134360907728775, 3.07939552385925686329295464339, 4.26936580159281048467713232171, 5.86167570820491382439076694934, 6.62915330582822778215013936478, 7.21821716447865331345990610727, 7.967477421025086437974286660438, 9.983196001486525948829947256632, 10.25079254161265134680590827031, 10.99022168541201805180365368170

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