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Properties

Label 2-507-39.2-c1-0-34
Degree $2$
Conductor $507$
Sign $-0.642 + 0.766i$
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  + (0.619 − 2.31i)2-s + (1.64 − 0.529i)3-s + (−3.23 − 1.86i)4-s + (1.69 + 1.69i)5-s + (−0.202 − 4.14i)6-s + (1.36 − 0.366i)7-s + (−2.93 + 2.93i)8-s + (2.43 − 1.74i)9-s + (4.96 − 2.86i)10-s + (−1.69 − 0.453i)11-s + (−6.31 − 1.36i)12-s − 3.38i·14-s + (3.68 + 1.89i)15-s + (1.23 + 2.13i)16-s + (−1.07 + 1.85i)17-s + (−2.52 − 6.72i)18-s + ⋯
L(s)  = 1  + (0.438 − 1.63i)2-s + (0.952 − 0.305i)3-s + (−1.61 − 0.933i)4-s + (0.757 + 0.757i)5-s + (−0.0826 − 1.69i)6-s + (0.516 − 0.138i)7-s + (−1.03 + 1.03i)8-s + (0.813 − 0.582i)9-s + (1.56 − 0.906i)10-s + (−0.510 − 0.136i)11-s + (−1.82 − 0.394i)12-s − 0.904i·14-s + (0.952 + 0.489i)15-s + (0.308 + 0.533i)16-s + (−0.260 + 0.450i)17-s + (−0.595 − 1.58i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08436 - 2.32476i\)
\(L(\frac12)\) \(\approx\) \(1.08436 - 2.32476i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 + 0.529i)T \)
13 \( 1 \)
good2 \( 1 + (-0.619 + 2.31i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + (-1.36 + 0.366i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.69 + 0.453i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.07 - 1.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.267 - i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.79 - 2.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.46 - 4.46i)T - 31iT^{2} \)
37 \( 1 + (-1.76 + 6.59i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.166 - 0.619i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.09 - 4.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (-1.23 - 4.62i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.46 - 2.26i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.62 + 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-6.09 - 6.09i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (9.70 + 2.60i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.36 + 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.76711712578824688547979554424, −9.868013591030098162484528497900, −9.219086687085310586451207518819, −8.107392332221407185430835025230, −7.00583069313306420897137090786, −5.66836629336547823930856055010, −4.36308546912651289176207392987, −3.31833748058406673864514898220, −2.42736156810364704307919461499, −1.55517106438764427463189720281, 2.11921062660247027095508527948, 3.87090041505874895280465307412, 4.97277409753283044878408069065, 5.42068457535407245861796797120, 6.72883844000573286779618515790, 7.71392367140448139630699517104, 8.329629940188430756196260887069, 9.174409080768675891859802830699, 9.783031085160026904467692600584, 11.17541018111125974572322737523

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