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Properties

Label 2-507-507.71-c1-0-34
Degree $2$
Conductor $507$
Sign $0.561 + 0.827i$
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  + (−1.64 + 0.548i)3-s + (1.44 − 1.38i)4-s + (0.680 − 0.0688i)7-s + (2.39 − 1.80i)9-s + (−1.60 + 3.06i)12-s + (−2.59 − 2.5i)13-s + (0.161 − 3.99i)16-s + (4.01 + 1.07i)19-s + (−1.08 + 0.486i)21-s + (4.11 − 2.84i)25-s + (−2.95 + 4.27i)27-s + (0.886 − 1.04i)28-s + (−1.98 − 10.8i)31-s + (0.962 − 5.92i)36-s + (3.49 − 9.81i)37-s + ⋯
L(s)  = 1  + (−0.948 + 0.316i)3-s + (0.721 − 0.692i)4-s + (0.257 − 0.0260i)7-s + (0.799 − 0.600i)9-s + (−0.464 + 0.885i)12-s + (−0.720 − 0.693i)13-s + (0.0402 − 0.999i)16-s + (0.922 + 0.247i)19-s + (−0.235 + 0.106i)21-s + (0.822 − 0.568i)25-s + (−0.568 + 0.822i)27-s + (0.167 − 0.197i)28-s + (−0.357 − 1.94i)31-s + (0.160 − 0.987i)36-s + (0.575 − 1.61i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07882 - 0.571726i\)
\(L(\frac12)\) \(\approx\) \(1.07882 - 0.571726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 - 0.548i)T \)
13 \( 1 + (2.59 + 2.5i)T \)
good2 \( 1 + (-1.44 + 1.38i)T^{2} \)
5 \( 1 + (-4.11 + 2.84i)T^{2} \)
7 \( 1 + (-0.680 + 0.0688i)T + (6.85 - 1.40i)T^{2} \)
11 \( 1 + (10.5 - 3.06i)T^{2} \)
17 \( 1 + (-3.40 - 16.6i)T^{2} \)
19 \( 1 + (-4.01 - 1.07i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-20.0 - 20.9i)T^{2} \)
31 \( 1 + (1.98 + 10.8i)T + (-28.9 + 10.9i)T^{2} \)
37 \( 1 + (-3.49 + 9.81i)T + (-28.6 - 23.4i)T^{2} \)
41 \( 1 + (24.6 + 32.7i)T^{2} \)
43 \( 1 + (-11.5 - 5.47i)T + (27.1 + 33.3i)T^{2} \)
47 \( 1 + (21.8 - 41.6i)T^{2} \)
53 \( 1 + (39.6 - 35.1i)T^{2} \)
59 \( 1 + (4.74 + 58.8i)T^{2} \)
61 \( 1 + (-11.3 + 1.84i)T + (57.8 - 19.3i)T^{2} \)
67 \( 1 + (-0.261 - 13.0i)T + (-66.9 + 2.69i)T^{2} \)
71 \( 1 + (27.8 - 65.3i)T^{2} \)
73 \( 1 + (13.1 - 10.3i)T + (17.4 - 70.8i)T^{2} \)
79 \( 1 + (-0.441 - 0.108i)T + (69.9 + 36.7i)T^{2} \)
83 \( 1 + (82.3 + 10.0i)T^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (16.5 - 3.73i)T + (87.6 - 41.5i)T^{2} \)
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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

−10.90651850561376879883751146675, −9.991460649656248185837354049047, −9.430709230941569196611771984203, −7.77880167829197018641647584674, −7.03795826120753537603822684259, −5.90225715350202615092296579988, −5.38407318778575331316375089936, −4.22495153733183067634279480663, −2.55066756299651012037954625714, −0.873860240257612464991770803839, 1.57930544662376958469431116986, 3.00983389064645492307147481291, 4.50449680874395553267883151784, 5.45972747774182800887286211324, 6.73351199404902903005627330240, 7.15135159348591272649134375770, 8.124334503257165048428133283598, 9.313904348947402090326485966213, 10.47501153154809226306437623612, 11.19521693525766781646296636507

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