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Properties

Label 2-507-13.10-c1-0-4
Degree $2$
Conductor $507$
Sign $-0.944 + 0.327i$
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  + (−1.77 + 1.02i)2-s + (0.5 + 0.866i)3-s + (1.09 − 1.90i)4-s + 3.35i·5-s + (−1.77 − 1.02i)6-s + (1.94 + 1.12i)7-s + 0.405i·8-s + (−0.499 + 0.866i)9-s + (−3.43 − 5.95i)10-s + (−4.27 + 2.46i)11-s + 2.19·12-s − 4.60·14-s + (−2.90 + 1.67i)15-s + (1.78 + 3.08i)16-s + (0.455 − 0.789i)17-s − 2.04i·18-s + ⋯
L(s)  = 1  + (−1.25 + 0.724i)2-s + (0.288 + 0.499i)3-s + (0.549 − 0.951i)4-s + 1.50i·5-s + (−0.724 − 0.418i)6-s + (0.735 + 0.424i)7-s + 0.143i·8-s + (−0.166 + 0.288i)9-s + (−1.08 − 1.88i)10-s + (−1.28 + 0.744i)11-s + 0.634·12-s − 1.23·14-s + (−0.750 + 0.433i)15-s + (0.445 + 0.771i)16-s + (0.110 − 0.191i)17-s − 0.482i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.944 + 0.327i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.114952 - 0.681747i\)
\(L(\frac12)\) \(\approx\) \(0.114952 - 0.681747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (1.77 - 1.02i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 3.35iT - 5T^{2} \)
7 \( 1 + (-1.94 - 1.12i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.27 - 2.46i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.455 + 0.789i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.29 - 1.90i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.01 - 1.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 - 3.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.82iT - 31T^{2} \)
37 \( 1 + (7.62 - 4.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.00 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.14 - 1.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 - 0.542T + 53T^{2} \)
59 \( 1 + (4.08 + 2.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.83 - 3.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.31 - 0.760i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.05 + 1.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.41iT - 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 2.30iT - 83T^{2} \)
89 \( 1 + (-8.71 + 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.9 - 8.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.89848054492272241118139620295, −10.27928292061870484997570873945, −9.696228988541895797044014778544, −8.646724201429354170525290449164, −7.63428222962658026842625426826, −7.38551782863934856907464156999, −6.13685833940453668644980781511, −5.03157317547601195081505123986, −3.39257626340778997560335807953, −2.16748650816633686982198396665, 0.61778297584089204469821026525, 1.58961213543626588221267076631, 2.96641589143885579295620926725, 4.73734372272029192425151700777, 5.56192139015211603365997862946, 7.37274206208676278693255798461, 8.140869535350351505225197968980, 8.591115265999338478399146728125, 9.353219369904167319192675457828, 10.42731170274068395992734656646

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