Properties

Label 2-507-39.20-c1-0-14
Degree $2$
Conductor $507$
Sign $0.884 - 0.466i$
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 + 1.5i)3-s + (1.73 − i)4-s + (4.23 + 1.13i)7-s + (−1.5 − 2.59i)9-s + 3.46i·12-s + (1.99 − 3.46i)16-s + (0.830 − 3.09i)19-s + (−5.36 + 5.36i)21-s + 5i·25-s + 5.19·27-s + (8.46 − 2.26i)28-s + (−0.830 − 0.830i)31-s + (−5.19 − 3i)36-s + (3.09 + 11.5i)37-s + (−1.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (1.59 + 0.428i)7-s + (−0.5 − 0.866i)9-s + 0.999i·12-s + (0.499 − 0.866i)16-s + (0.190 − 0.710i)19-s + (−1.17 + 1.17i)21-s + i·25-s + 1.00·27-s + (1.59 − 0.428i)28-s + (−0.149 − 0.149i)31-s + (−0.866 − 0.5i)36-s + (0.509 + 1.90i)37-s + (−0.228 + 0.132i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.884 - 0.466i$
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: \((2,\ 507,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66188 + 0.411084i\)
\(L(\frac12)\) \(\approx\) \(1.66188 + 0.411084i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + i)T^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (-4.23 - 1.13i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.830 + 3.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.830 + 0.830i)T + 31iT^{2} \)
37 \( 1 + (-3.09 - 11.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.5 - 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.33 + 7.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (15.7 - 4.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-7.63 + 7.63i)T - 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.57 + 9.59i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.20925615859873701658982784383, −10.28240227607318167173007411621, −9.356833302317677016558555965150, −8.355331293605920378719470314057, −7.29796146856845423946532991222, −6.15074601469211428034605277285, −5.25939419176201512386259936420, −4.59688469167646466516192738969, −2.96815786718822179093123457294, −1.48776120123247989829559681577, 1.40453528615404215089760891072, 2.42371964616249924466544536967, 4.12378034154647058043967478489, 5.36430651755758200369668436988, 6.32926387454556040544914195672, 7.43157918959302195069625635514, 7.80534587103804778134967012281, 8.672150241457403921887812192522, 10.45399637184868832001601321914, 10.97186613367490808747548052631

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