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Properties

Label 2-507-13.4-c1-0-18
Degree $2$
Conductor $507$
Sign $0.702 + 0.711i$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (−0.500 − 0.866i)4-s + i·5-s + (0.866 − 0.499i)6-s + (1.73 − i)7-s − 3i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (1.73 + i)11-s − 12-s + 1.99·14-s + (0.866 + 0.5i)15-s + (0.500 − 0.866i)16-s + (−3.5 − 6.06i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (−0.250 − 0.433i)4-s + 0.447i·5-s + (0.353 − 0.204i)6-s + (0.654 − 0.377i)7-s − 1.06i·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.522 + 0.301i)11-s − 0.288·12-s + 0.534·14-s + (0.223 + 0.129i)15-s + (0.125 − 0.216i)16-s + (−0.848 − 1.47i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92805 - 0.805877i\)
\(L(\frac12)\) \(\approx\) \(1.92805 - 0.805877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T + (1 + 1.73i)T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.19 + 3i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.79 - 4.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.19 - 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + (-12.1 - 7i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.05011962081984053923244774411, −9.635589571092734545916059409684, −9.258684334143569045590373565015, −7.72990013625310913527757562497, −7.10852741459749252688015635459, −6.23463890766177822734991122110, −5.07683216000238091358954193731, −4.23561651004958191088107881738, −2.84200221616373123076709201883, −1.15376585142794175515460491807, 1.93980892458886196111240488829, 3.34110204383868907799501070667, 4.25679356984428920456548589490, 5.04669575786185700694298464037, 6.09692496630906208277571074719, 7.67527142596322194971625883130, 8.666390930532732156270877228552, 8.861322492459847292989871687916, 10.31780666974811209430989113822, 11.12061451207057110526474500947

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