Properties

Label 2-507-169.153-c1-0-0
Degree $2$
Conductor $507$
Sign $0.0997 + 0.995i$
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  + (−0.157 + 1.95i)2-s + (−0.845 − 0.534i)3-s + (−1.83 − 0.297i)4-s + (−2.06 + 2.32i)5-s + (1.17 − 1.56i)6-s + (−0.626 − 0.601i)7-s + (−0.0680 + 0.276i)8-s + (0.428 + 0.903i)9-s + (−4.23 − 4.40i)10-s + (1.11 + 0.531i)11-s + (1.38 + 1.23i)12-s + (−3.59 − 0.238i)13-s + (1.27 − 1.13i)14-s + (2.98 − 0.865i)15-s + (−4.04 − 1.35i)16-s + (2.79 − 2.91i)17-s + ⋯
L(s)  = 1  + (−0.111 + 1.38i)2-s + (−0.487 − 0.308i)3-s + (−0.915 − 0.148i)4-s + (−0.922 + 1.04i)5-s + (0.481 − 0.640i)6-s + (−0.236 − 0.227i)7-s + (−0.0240 + 0.0976i)8-s + (0.142 + 0.301i)9-s + (−1.33 − 1.39i)10-s + (0.337 + 0.160i)11-s + (0.400 + 0.355i)12-s + (−0.997 − 0.0661i)13-s + (0.341 − 0.302i)14-s + (0.771 − 0.223i)15-s + (−1.01 − 0.337i)16-s + (0.678 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0997 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0997 + 0.995i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0997 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117707 - 0.106495i\)
\(L(\frac12)\) \(\approx\) \(0.117707 - 0.106495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.845 + 0.534i)T \)
13 \( 1 + (3.59 + 0.238i)T \)
good2 \( 1 + (0.157 - 1.95i)T + (-1.97 - 0.320i)T^{2} \)
5 \( 1 + (2.06 - 2.32i)T + (-0.602 - 4.96i)T^{2} \)
7 \( 1 + (0.626 + 0.601i)T + (0.281 + 6.99i)T^{2} \)
11 \( 1 + (-1.11 - 0.531i)T + (6.95 + 8.52i)T^{2} \)
17 \( 1 + (-2.79 + 2.91i)T + (-0.684 - 16.9i)T^{2} \)
19 \( 1 + (2.44 + 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.19 + 2.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.53 + 0.204i)T + (28.6 + 4.65i)T^{2} \)
31 \( 1 + (-1.31 - 0.159i)T + (30.0 + 7.41i)T^{2} \)
37 \( 1 + (2.78 + 6.54i)T + (-25.6 + 26.6i)T^{2} \)
41 \( 1 + (3.02 - 4.78i)T + (-17.5 - 37.0i)T^{2} \)
43 \( 1 + (2.78 + 1.18i)T + (29.7 + 31.0i)T^{2} \)
47 \( 1 + (-4.83 - 1.83i)T + (35.1 + 31.1i)T^{2} \)
53 \( 1 + (9.34 + 2.30i)T + (46.9 + 24.6i)T^{2} \)
59 \( 1 + (-4.77 - 14.2i)T + (-47.1 + 35.4i)T^{2} \)
61 \( 1 + (3.99 - 13.7i)T + (-51.5 - 32.6i)T^{2} \)
67 \( 1 + (0.0467 + 0.287i)T + (-63.5 + 21.2i)T^{2} \)
71 \( 1 + (7.55 + 0.304i)T + (70.7 + 5.71i)T^{2} \)
73 \( 1 + (-1.45 + 1.00i)T + (25.8 - 68.2i)T^{2} \)
79 \( 1 + (1.23 - 3.25i)T + (-59.1 - 52.3i)T^{2} \)
83 \( 1 + (6.34 - 12.0i)T + (-47.1 - 68.3i)T^{2} \)
89 \( 1 + (5.23 - 3.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.09 + 1.03i)T + (89.2 - 38.0i)T^{2} \)
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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

−11.62947084325026456060377205457, −10.73937314085764463720027065992, −9.750806197680672645148093583068, −8.532716844684074645010423084392, −7.43477938391469292816059845436, −7.21666810910799402339310287225, −6.40136994454571087698599570067, −5.33899167913433591455906105034, −4.20368468713715228930824442153, −2.70265577057971793980556869897, 0.10291277029948207658978148449, 1.61994857205256095178668465005, 3.31383011693233905666168913362, 4.18041072257922140367460259107, 5.08004295312170288484543740416, 6.41057637049554434600573998019, 7.78318263819666023803515733251, 8.762712773705725751336621660486, 9.606873871476448931171787972427, 10.31352188911507631395843215730

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