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Properties

Label 2-507-39.11-c1-0-9
Degree $2$
Conductor $507$
Sign $0.0257 - 0.999i$
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  + (−2.31 − 0.619i)2-s + (0.866 + 1.5i)3-s + (3.23 + 1.86i)4-s + (1.23 − 1.23i)5-s + (−1.07 − 4.00i)6-s + (−2.93 − 2.93i)8-s + (−1.5 + 2.59i)9-s + (−3.63 + 2.09i)10-s + (−1.69 + 6.31i)11-s + 6.46i·12-s + (2.93 + 0.785i)15-s + (1.23 + 2.13i)16-s + (5.07 − 5.07i)18-s + (6.31 − 1.69i)20-s + (7.83 − 13.5i)22-s + ⋯
L(s)  = 1  + (−1.63 − 0.438i)2-s + (0.499 + 0.866i)3-s + (1.61 + 0.933i)4-s + (0.554 − 0.554i)5-s + (−0.438 − 1.63i)6-s + (−1.03 − 1.03i)8-s + (−0.5 + 0.866i)9-s + (−1.14 + 0.663i)10-s + (−0.510 + 1.90i)11-s + 1.86i·12-s + (0.757 + 0.202i)15-s + (0.308 + 0.533i)16-s + (1.19 − 1.19i)18-s + (1.41 − 0.378i)20-s + (1.66 − 2.89i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0257 - 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0257 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.495340 + 0.482727i\)
\(L(\frac12)\) \(\approx\) \(0.495340 + 0.482727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (2.31 + 0.619i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-1.23 + 1.23i)T - 5iT^{2} \)
7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.69 - 6.31i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.55 + 2.02i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.10 - 7.10i)T + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.453 - 0.121i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.92 - 12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.17 - 15.5i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-8.91 + 8.91i)T - 83iT^{2} \)
89 \( 1 + (-1.11 + 4.17i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (84.0 - 48.5i)T^{2} \)
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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

−10.53134967143072213107300360298, −10.11352346528667162096331063102, −9.367864581697799698799726205056, −8.865688966011600301737701961520, −7.83481588289724379641211394639, −7.11428451712219229494431767466, −5.39911689280771233709548016436, −4.35260265614653042536752890844, −2.67917700727704387932272091423, −1.72473322487527785756129841546, 0.64783047454391496623114860193, 2.12727199003770970708159030772, 3.24332505869144330039028124369, 5.77720632788032225436784804712, 6.38428031119919749248289584911, 7.27845914853163933064098600113, 8.187906488405922463497791296399, 8.678525895525545400628750333422, 9.560989531446916500694780071204, 10.56853990909347140640761900761

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