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Properties

Label 2-507-13.10-c1-0-14
Degree $2$
Conductor $507$
Sign $0.543 + 0.839i$
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  + (−2.09 + 1.20i)2-s + (−0.5 − 0.866i)3-s + (1.91 − 3.31i)4-s + 2.82i·5-s + (2.09 + 1.20i)6-s + (−2.44 − 1.41i)7-s + 4.41i·8-s + (−0.499 + 0.866i)9-s + (−3.41 − 5.91i)10-s + (1.73 − i)11-s − 3.82·12-s + 6.82·14-s + (2.44 − 1.41i)15-s + (−1.49 − 2.59i)16-s + (−1.82 + 3.16i)17-s − 2.41i·18-s + ⋯
L(s)  = 1  + (−1.47 + 0.853i)2-s + (−0.288 − 0.499i)3-s + (0.957 − 1.65i)4-s + 1.26i·5-s + (0.853 + 0.492i)6-s + (−0.925 − 0.534i)7-s + 1.56i·8-s + (−0.166 + 0.288i)9-s + (−1.07 − 1.87i)10-s + (0.522 − 0.301i)11-s − 1.10·12-s + 1.82·14-s + (0.632 − 0.365i)15-s + (−0.374 − 0.649i)16-s + (−0.443 + 0.768i)17-s − 0.569i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.543 + 0.839i$
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: \((2,\ 507,\ (\ :1/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267327 - 0.145281i\)
\(L(\frac12)\) \(\approx\) \(0.267327 - 0.145281i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 1.20i)T + (1 - 1.73i)T^{2} \)
5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 + (2.44 + 1.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.44 + 1.41i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 + (3.16 - 1.82i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.37 + 5.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.82 + 8.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (3.16 + 1.82i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.65 + 8.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.01 + 0.585i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 + i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 7.65iT - 83T^{2} \)
89 \( 1 + (7.94 - 4.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.63 - 3.82i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.68764771426663374243025677105, −9.825961310438446507029609147637, −8.933044987290379555024597949874, −7.911323395852361971722653661617, −7.08322289919528346998934200924, −6.47080818618602969500106200429, −6.01306850867103621189615382580, −3.83190686593396048716901535691, −2.22910552121867770650686031409, −0.33203682507919674129743592743, 1.24987168695827252383152633048, 2.76772849945399371477076706987, 4.11483655770346194059593118434, 5.38103908968728873703769167642, 6.64574130143710415682398055558, 7.898527308866619456593038273831, 8.946535444596203297608430440394, 9.242386551667084966960036030862, 9.898468172832949931926505364361, 10.90052631610723321565222783586

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