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Properties

Label 2-507-39.2-c1-0-11
Degree $2$
Conductor $507$
Sign $0.268 - 0.963i$
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 + (−1.28 + 1.16i)3-s + (−3.23 − 1.86i)4-s + (−1.69 − 1.69i)5-s + (−1.89 − 3.68i)6-s + (1.36 − 0.366i)7-s + (2.93 − 2.93i)8-s + (0.292 − 2.98i)9-s + (4.96 − 2.86i)10-s + (1.69 + 0.453i)11-s + (6.31 − 1.36i)12-s + 3.38i·14-s + (4.14 + 0.202i)15-s + (1.23 + 2.13i)16-s + (1.07 − 1.85i)17-s + (6.72 + 2.52i)18-s + ⋯
L(s)  = 1  + (−0.438 + 1.63i)2-s + (−0.740 + 0.671i)3-s + (−1.61 − 0.933i)4-s + (−0.757 − 0.757i)5-s + (−0.773 − 1.50i)6-s + (0.516 − 0.138i)7-s + (1.03 − 1.03i)8-s + (0.0975 − 0.995i)9-s + (1.56 − 0.906i)10-s + (0.510 + 0.136i)11-s + (1.82 − 0.394i)12-s + 0.904i·14-s + (1.06 + 0.0523i)15-s + (0.308 + 0.533i)16-s + (0.260 − 0.450i)17-s + (1.58 + 0.595i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.268 - 0.963i$
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: \((2,\ 507,\ (\ :1/2),\ 0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.532046 + 0.403935i\)
\(L(\frac12)\) \(\approx\) \(0.532046 + 0.403935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 - 1.16i)T \)
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

−11.02799179097286943065414696313, −9.864321376018370397016787287596, −9.086848917704843004949066397024, −8.328670341306335446662152313018, −7.46019663829468096226655595068, −6.54363033783400013894503016156, −5.51067410497168262336229020497, −4.77898448597681675133552923683, −3.96024065086321508824707359009, −0.67182908395709811945723892526, 1.07502410790186060628515781433, 2.37419873775286964796565961732, 3.58302209362820257260827009076, 4.68028899608053317726174270188, 6.13043879799524928349476195560, 7.29852028586964167182892325580, 8.130242298340361234306026746810, 9.132131137258583561502246019587, 10.33911899773113299386503566389, 10.92854689729093374876693264787

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