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Properties

Label 2-507-39.32-c1-0-2
Degree $2$
Conductor $507$
Sign $-0.0257 - 0.999i$
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  + (−1.45 + 0.389i)2-s + (−0.866 + 1.5i)3-s + (0.232 − 0.133i)4-s + (−2.90 − 2.90i)5-s + (0.675 − 2.51i)6-s + (1.84 − 1.84i)8-s + (−1.5 − 2.59i)9-s + (5.36 + 3.09i)10-s + (0.285 + 1.06i)11-s + 0.464i·12-s + (6.88 − 1.84i)15-s + (−2.23 + 3.86i)16-s + (3.19 + 3.19i)18-s + (−1.06 − 0.285i)20-s + (−0.830 − 1.43i)22-s + ⋯
L(s)  = 1  + (−1.02 + 0.275i)2-s + (−0.499 + 0.866i)3-s + (0.116 − 0.0669i)4-s + (−1.30 − 1.30i)5-s + (0.275 − 1.02i)6-s + (0.652 − 0.652i)8-s + (−0.5 − 0.866i)9-s + (1.69 + 0.979i)10-s + (0.0860 + 0.321i)11-s + 0.133i·12-s + (1.77 − 0.476i)15-s + (−0.558 + 0.966i)16-s + (0.752 + 0.752i)18-s + (−0.238 − 0.0638i)20-s + (−0.176 − 0.306i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0257 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.0257 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.0257 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.232124 + 0.238189i\)
\(L(\frac12)\) \(\approx\) \(0.232124 + 0.238189i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.389i)T + (1.73 - i)T^{2} \)
5 \( 1 + (2.90 + 2.90i)T + 5iT^{2} \)
7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.285 - 1.06i)T + (-9.52 + 5.5i)T^{2} \)
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 \( 1 + (9.79 - 2.62i)T + (35.5 - 20.5i)T^{2} \)
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 \( 1 + (-14.8 - 3.97i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.92 - 12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.27 + 4.75i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (-9.29 - 9.29i)T + 83iT^{2} \)
89 \( 1 + (-4.75 - 17.7i)T + (-77.0 + 44.5i)T^{2} \)
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

−11.01022345875017576285753709779, −10.01984663442921860140234684732, −9.220118815268147238460570935868, −8.584644007604092618509876287980, −7.87733803502738287476333798777, −6.83604617549782154748949369709, −5.28961122031086998864488871248, −4.42866599104787517010859463672, −3.71926510659350114498748403702, −0.902265000529904177501658699656, 0.43033674998247893913707154931, 2.21504355745170351523548726950, 3.58277553856310040786373012310, 5.08638648359128505168015203566, 6.50508542082357996420052085290, 7.22557547601390549317950480509, 7.997652445725386643738513560432, 8.627812112238781376259682790704, 10.08896167510063234000513589268, 10.72521775039234083652157345497

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