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Properties

Label 2-507-39.2-c1-0-38
Degree $2$
Conductor $507$
Sign $-0.533 - 0.846i$
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  + (0.389 − 1.45i)2-s + (−0.866 − 1.5i)3-s + (−0.232 − 0.133i)4-s + (−2.90 − 2.90i)5-s + (−2.51 + 0.675i)6-s + (1.84 − 1.84i)8-s + (−1.5 + 2.59i)9-s + (−5.36 + 3.09i)10-s + (−1.06 − 0.285i)11-s + 0.464i·12-s + (−1.84 + 6.88i)15-s + (−2.23 − 3.86i)16-s + (3.19 + 3.19i)18-s + (0.285 + 1.06i)20-s + (−0.830 + 1.43i)22-s + ⋯
L(s)  = 1  + (0.275 − 1.02i)2-s + (−0.499 − 0.866i)3-s + (−0.116 − 0.0669i)4-s + (−1.30 − 1.30i)5-s + (−1.02 + 0.275i)6-s + (0.652 − 0.652i)8-s + (−0.5 + 0.866i)9-s + (−1.69 + 0.979i)10-s + (−0.321 − 0.0860i)11-s + 0.133i·12-s + (−0.476 + 1.77i)15-s + (−0.558 − 0.966i)16-s + (0.752 + 0.752i)18-s + (0.0638 + 0.238i)20-s + (−0.176 + 0.306i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.533 - 0.846i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.533 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407783 + 0.738888i\)
\(L(\frac12)\) \(\approx\) \(0.407783 + 0.738888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.389 + 1.45i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (2.90 + 2.90i)T + 5iT^{2} \)
7 \( 1 + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.06 + 0.285i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.62 + 9.79i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.59 - 6.59i)T - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (3.97 + 14.8i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.92 + 12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.75 - 1.27i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-9.29 - 9.29i)T + 83iT^{2} \)
89 \( 1 + (17.7 + 4.75i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-84.0 + 48.5i)T^{2} \)
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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

−10.91173815584280209781114506104, −9.541156529431912016578264630718, −8.313021387241324470700621438967, −7.75240836376866608702198848389, −6.78304921080310488546844998512, −5.28906367053479571257649717561, −4.44691056243219158394704734435, −3.30848896402607797933742085126, −1.76113247808402010973505500028, −0.48634814710896855157750741463, 2.91522817695959817203080778591, 4.03409763126126872004764144372, 4.95338289364915345740152474156, 6.12252217093397613492932112369, 6.82976325543520716425656325428, 7.66644114918838644346555336856, 8.495199275477907060240920566752, 9.990077169939600508914228775188, 10.69920591405193129139486670476, 11.38063440062360944667527971963

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