Properties

Label 2-507-39.32-c1-0-19
Degree $2$
Conductor $507$
Sign $0.281 - 0.959i$
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  + (−2.31 + 0.619i)2-s + (1.64 + 0.529i)3-s + (3.23 − 1.86i)4-s + (1.69 + 1.69i)5-s + (−4.14 − 0.202i)6-s + (0.366 − 1.36i)7-s + (−2.93 + 2.93i)8-s + (2.43 + 1.74i)9-s + (−4.96 − 2.86i)10-s + (0.453 + 1.69i)11-s + (6.31 − 1.36i)12-s + 3.38i·14-s + (1.89 + 3.68i)15-s + (1.23 − 2.13i)16-s + (1.07 + 1.85i)17-s + (−6.72 − 2.52i)18-s + ⋯
L(s)  = 1  + (−1.63 + 0.438i)2-s + (0.952 + 0.305i)3-s + (1.61 − 0.933i)4-s + (0.757 + 0.757i)5-s + (−1.69 − 0.0826i)6-s + (0.138 − 0.516i)7-s + (−1.03 + 1.03i)8-s + (0.813 + 0.582i)9-s + (−1.56 − 0.906i)10-s + (0.136 + 0.510i)11-s + (1.82 − 0.394i)12-s + 0.904i·14-s + (0.489 + 0.952i)15-s + (0.308 − 0.533i)16-s + (0.260 + 0.450i)17-s + (−1.58 − 0.595i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875005 + 0.655513i\)
\(L(\frac12)\) \(\approx\) \(0.875005 + 0.655513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.529i)T \)
13 \( 1 \)
good2 \( 1 + (2.31 - 0.619i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + (-0.366 + 1.36i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.453 - 1.69i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.07 - 1.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 0.267i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.79 + 2.76i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 + 4.46i)T - 31iT^{2} \)
37 \( 1 + (-6.59 + 1.76i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.619 + 0.166i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.09 - 4.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 + 4.62iT - 53T^{2} \)
59 \( 1 + (4.62 + 1.23i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 - 8.46i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.23 - 4.62i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.09 + 6.09i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-2.60 - 9.70i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.5 + 3.36i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.56191761866180207428615104429, −9.799148065024493174149142799066, −9.583087327149695266304669654871, −8.381744172806616499897745075253, −7.70139041522317021765024169660, −6.94290116452962223740342052681, −5.99635707702748889758638225144, −4.23802177462681480915676742027, −2.67892749542393242998478622004, −1.58295154897055578782395973987, 1.14112696066653725154897812640, 2.12763078534259849622766685025, 3.25818926469442174774571190645, 5.12590690739692516227907041671, 6.51227470848891205098105328816, 7.57510087254043119625147833433, 8.418236015789392275949493728634, 8.987903023908555265873712004541, 9.551064831666355912790982314254, 10.30632358353984924217587349516

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