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Properties

Label 2-507-39.20-c1-0-9
Degree $2$
Conductor $507$
Sign $-0.898 - 0.438i$
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  + (0.506 + 1.89i)2-s + (−1.69 − 0.369i)3-s + (−1.58 + 0.915i)4-s + (−1.04 + 1.04i)5-s + (−0.159 − 3.38i)6-s + (4.33 + 1.16i)7-s + (0.233 + 0.233i)8-s + (2.72 + 1.24i)9-s + (−2.50 − 1.44i)10-s + (0.147 − 0.0396i)11-s + (3.02 − 0.964i)12-s + 8.77i·14-s + (2.15 − 1.38i)15-s + (−2.15 + 3.73i)16-s + (−1.58 − 2.74i)17-s + (−0.980 + 5.78i)18-s + ⋯
L(s)  = 1  + (0.358 + 1.33i)2-s + (−0.977 − 0.213i)3-s + (−0.792 + 0.457i)4-s + (−0.468 + 0.468i)5-s + (−0.0650 − 1.38i)6-s + (1.63 + 0.438i)7-s + (0.0825 + 0.0825i)8-s + (0.909 + 0.416i)9-s + (−0.793 − 0.458i)10-s + (0.0446 − 0.0119i)11-s + (0.872 − 0.278i)12-s + 2.34i·14-s + (0.557 − 0.357i)15-s + (−0.538 + 0.932i)16-s + (−0.384 − 0.665i)17-s + (−0.231 + 1.36i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.898 - 0.438i$
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: \((2,\ 507,\ (\ :1/2),\ -0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299511 + 1.29642i\)
\(L(\frac12)\) \(\approx\) \(0.299511 + 1.29642i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.369i)T \)
13 \( 1 \)
good2 \( 1 + (-0.506 - 1.89i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.04 - 1.04i)T - 5iT^{2} \)
7 \( 1 + (-4.33 - 1.16i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.147 + 0.0396i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.58 + 2.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.309 - 1.15i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.35 - 5.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.71 - 0.992i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.64 + 3.64i)T + 31iT^{2} \)
37 \( 1 + (0.847 + 3.16i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.16 - 8.08i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.41 + 1.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.06 - 4.06i)T + 47iT^{2} \)
53 \( 1 + 0.628iT - 53T^{2} \)
59 \( 1 + (-2.27 + 8.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.91 + 5.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.649 + 0.174i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-11.0 - 2.96i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.92 + 3.92i)T - 73iT^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 + (-6.40 + 6.40i)T - 83iT^{2} \)
89 \( 1 + (-5.12 + 1.37i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.15 - 4.32i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.29025432437958257492329044747, −10.83270281899324105994056169316, −9.308998356465707866919749830132, −7.934181285389176079965879861484, −7.65873455428794804742061635399, −6.69537470680635178310975430135, −5.66687265825198787516477486786, −5.07157701977936500589084277165, −4.13293537528145632649195583996, −1.84107659072445889327529251479, 0.863810256706189152924906248456, 2.07005436215312060604591302588, 4.04961055633737318344610644184, 4.42729577671756988743846827177, 5.34414562734066744168366156720, 6.81399478780457092467299945370, 7.920448956977513422895784900271, 8.908624790162493626595437187643, 10.35239641912884769465143831592, 10.66415715226955963684311691608

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