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Properties

Label 2-507-39.8-c1-0-28
Degree $2$
Conductor $507$
Sign $0.942 + 0.333i$
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  + (−1.82 + 1.82i)2-s + (1.43 − 0.964i)3-s − 4.66i·4-s + (0.624 − 0.624i)5-s + (−0.865 + 4.38i)6-s + (1.18 − 1.18i)7-s + (4.86 + 4.86i)8-s + (1.13 − 2.77i)9-s + 2.28i·10-s + (−0.253 − 0.253i)11-s + (−4.49 − 6.70i)12-s + 4.34i·14-s + (0.296 − 1.50i)15-s − 8.41·16-s − 2.62·17-s + (2.98 + 7.14i)18-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)2-s + (0.830 − 0.556i)3-s − 2.33i·4-s + (0.279 − 0.279i)5-s + (−0.353 + 1.79i)6-s + (0.449 − 0.449i)7-s + (1.71 + 1.71i)8-s + (0.379 − 0.925i)9-s + 0.721i·10-s + (−0.0763 − 0.0763i)11-s + (−1.29 − 1.93i)12-s + 1.15i·14-s + (0.0764 − 0.387i)15-s − 2.10·16-s − 0.637·17-s + (0.703 + 1.68i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978313 - 0.167759i\)
\(L(\frac12)\) \(\approx\) \(0.978313 - 0.167759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.43 + 0.964i)T \)
13 \( 1 \)
good2 \( 1 + (1.82 - 1.82i)T - 2iT^{2} \)
5 \( 1 + (-0.624 + 0.624i)T - 5iT^{2} \)
7 \( 1 + (-1.18 + 1.18i)T - 7iT^{2} \)
11 \( 1 + (0.253 + 0.253i)T + 11iT^{2} \)
17 \( 1 + 2.62T + 17T^{2} \)
19 \( 1 + (4.32 + 4.32i)T + 19iT^{2} \)
23 \( 1 - 5.41T + 23T^{2} \)
29 \( 1 + 8.37iT - 29T^{2} \)
31 \( 1 + (1.27 + 1.27i)T + 31iT^{2} \)
37 \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \)
41 \( 1 + (-4.89 + 4.89i)T - 41iT^{2} \)
43 \( 1 + 0.952iT - 43T^{2} \)
47 \( 1 + (5.33 + 5.33i)T + 47iT^{2} \)
53 \( 1 - 9.69iT - 53T^{2} \)
59 \( 1 + (-7.28 - 7.28i)T + 59iT^{2} \)
61 \( 1 - 4.87T + 61T^{2} \)
67 \( 1 + (-8.90 - 8.90i)T + 67iT^{2} \)
71 \( 1 + (2.27 - 2.27i)T - 71iT^{2} \)
73 \( 1 + (0.246 - 0.246i)T - 73iT^{2} \)
79 \( 1 + 3.68T + 79T^{2} \)
83 \( 1 + (-8.47 + 8.47i)T - 83iT^{2} \)
89 \( 1 + (-4.84 - 4.84i)T + 89iT^{2} \)
97 \( 1 + (3.74 + 3.74i)T + 97iT^{2} \)
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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

−10.51612349696437546867612161604, −9.418468161476573030317119901334, −8.940736671622182522724266662716, −8.177871346125944834331642600934, −7.33614279308365838881661769125, −6.74068830767937279379891114127, −5.68039270743577128100210859728, −4.35827768790345469774160702782, −2.25836719286572880115766396572, −0.856701833944888696543258826968, 1.74656751217786875001632400688, 2.62394146360888111340849862306, 3.65401914696291951216309327317, 4.88742209773313505310517769983, 6.76428663493894782777914159415, 8.040375816130580778544901868363, 8.507849220650181858223033764529, 9.301886301133037043647213915652, 10.04523322672250173649083073134, 10.79284916415321222063429948256

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