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Properties

Label 2-507-39.5-c1-0-10
Degree $2$
Conductor $507$
Sign $-0.980 - 0.198i$
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  + (1.69 + 1.69i)2-s + (−0.366 + 1.69i)3-s + 3.73i·4-s + (1.69 + 1.69i)5-s + (−3.48 + 2.24i)6-s + (−1 − i)7-s + (−2.93 + 2.93i)8-s + (−2.73 − 1.23i)9-s + 5.73i·10-s + (1.23 − 1.23i)11-s + (−6.31 − 1.36i)12-s − 3.38i·14-s + (−3.48 + 2.24i)15-s − 2.46·16-s + 2.14·17-s + (−2.52 − 6.72i)18-s + ⋯
L(s)  = 1  + (1.19 + 1.19i)2-s + (−0.211 + 0.977i)3-s + 1.86i·4-s + (0.757 + 0.757i)5-s + (−1.42 + 0.917i)6-s + (−0.377 − 0.377i)7-s + (−1.03 + 1.03i)8-s + (−0.910 − 0.413i)9-s + 1.81i·10-s + (0.373 − 0.373i)11-s + (−1.82 − 0.394i)12-s − 0.904i·14-s + (−0.899 + 0.580i)15-s − 0.616·16-s + 0.520·17-s + (−0.595 − 1.58i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.980 - 0.198i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.980 - 0.198i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262972 + 2.62143i\)
\(L(\frac12)\) \(\approx\) \(0.262972 + 2.62143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.366 - 1.69i)T \)
13 \( 1 \)
good2 \( 1 + (-1.69 - 1.69i)T + 2iT^{2} \)
5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (-1.23 + 1.23i)T - 11iT^{2} \)
17 \( 1 - 2.14T + 17T^{2} \)
19 \( 1 + (-0.732 + 0.732i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 5.53iT - 29T^{2} \)
31 \( 1 + (4.46 - 4.46i)T - 31iT^{2} \)
37 \( 1 + (-4.83 - 4.83i)T + 37iT^{2} \)
41 \( 1 + (0.453 + 0.453i)T + 41iT^{2} \)
43 \( 1 + 8.19iT - 43T^{2} \)
47 \( 1 + (6.77 - 6.77i)T - 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (-3.38 + 3.38i)T - 59iT^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (6.19 - 6.19i)T - 67iT^{2} \)
71 \( 1 + (3.38 + 3.38i)T + 71iT^{2} \)
73 \( 1 + (-6.09 - 6.09i)T + 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 + 1.23i)T + 83iT^{2} \)
89 \( 1 + (-7.10 + 7.10i)T - 89iT^{2} \)
97 \( 1 + (9.19 - 9.19i)T - 97iT^{2} \)
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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

−11.37301481682894960287201719178, −10.35834460691950569740070693658, −9.666571579510559445786253536440, −8.501862320002860941864392624401, −7.30727389013985837338323838891, −6.35643181333625865873594855432, −5.85614244559148914101319139811, −4.85107761994723456363776783621, −3.80138962565327556255201378034, −2.97219060780929974974974115437, 1.27522687636798636771967110223, 2.19243624944762608849094562519, 3.39080671197382563548933872231, 4.83577485719117528973645449524, 5.61646917153109894097654741854, 6.32112100644091403783122983196, 7.67226141347859155148868675635, 9.031350613132712580229712917305, 9.763061502531668235877935342003, 10.86725014741183431811988811400

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