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Properties

Label 2-507-39.20-c1-0-19
Degree $2$
Conductor $507$
Sign $0.603 - 0.796i$
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  + (0.258 + 0.965i)2-s + (1.72 − 0.158i)3-s + (0.866 − 0.5i)4-s + (−1.41 + 1.41i)5-s + (0.599 + 1.62i)6-s + (−1.36 − 0.366i)7-s + (2.12 + 2.12i)8-s + (2.94 − 0.548i)9-s + (−1.73 − 1.00i)10-s + (3.86 − 1.03i)11-s + (1.41 − i)12-s − 1.41i·14-s + (−2.21 + 2.66i)15-s + (−0.500 + 0.866i)16-s + (1.29 + 2.70i)18-s + (0.366 − 1.36i)19-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (0.995 − 0.0917i)3-s + (0.433 − 0.250i)4-s + (−0.632 + 0.632i)5-s + (0.244 + 0.663i)6-s + (−0.516 − 0.138i)7-s + (0.749 + 0.749i)8-s + (0.983 − 0.182i)9-s + (−0.547 − 0.316i)10-s + (1.16 − 0.312i)11-s + (0.408 − 0.288i)12-s − 0.377i·14-s + (−0.571 + 0.687i)15-s + (−0.125 + 0.216i)16-s + (0.304 + 0.638i)18-s + (0.0839 − 0.313i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.603 - 0.796i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.603 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08915 + 1.03807i\)
\(L(\frac12)\) \(\approx\) \(2.08915 + 1.03807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.158i)T \)
13 \( 1 \)
good2 \( 1 + (-0.258 - 0.965i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.86 + 1.03i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.366 + 1.36i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.24 - 7.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.44 + 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.517 + 1.93i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.19 + 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (1.03 - 3.86i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.86 + 1.03i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (5.65 - 5.65i)T - 83iT^{2} \)
89 \( 1 + (13.5 - 3.62i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.56 + 9.56i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.16647613096733762806341880125, −9.969728009063901102184436006488, −9.224332255261250819216675729805, −8.055848202729948899275024416273, −7.31608089031070367214673912032, −6.73356550181141798691709595515, −5.69986478752132461204150850726, −4.06306649563595096426307907961, −3.29195401144391934787623422629, −1.81845394087668786338812573208, 1.51354470790136810273353816060, 2.80286239047113522615924923616, 3.87162640086216596868383671015, 4.44487677358451694820468708561, 6.37460174038937137824029962775, 7.28319285632328771226744697297, 8.201818576258523964269147405279, 9.038365759802183692315536725774, 9.881095272517331749449160542019, 10.78456876618028435805385698058

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