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Properties

Label 2-507-13.4-c1-0-22
Degree $2$
Conductor $507$
Sign $0.702 + 0.711i$
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  + (1.35 + 0.780i)2-s + (−0.5 + 0.866i)3-s + (0.219 + 0.379i)4-s − 3.56i·5-s + (−1.35 + 0.780i)6-s + (−0.486 + 0.280i)7-s − 2.43i·8-s + (−0.499 − 0.866i)9-s + (2.78 − 4.81i)10-s + (−1.73 − i)11-s − 0.438·12-s − 0.876·14-s + (3.08 + 1.78i)15-s + (2.34 − 4.05i)16-s + (−0.780 − 1.35i)17-s − 1.56i·18-s + ⋯
L(s)  = 1  + (0.956 + 0.552i)2-s + (−0.288 + 0.499i)3-s + (0.109 + 0.189i)4-s − 1.59i·5-s + (−0.552 + 0.318i)6-s + (−0.183 + 0.106i)7-s − 0.862i·8-s + (−0.166 − 0.288i)9-s + (0.879 − 1.52i)10-s + (−0.522 − 0.301i)11-s − 0.126·12-s − 0.234·14-s + (0.796 + 0.459i)15-s + (0.585 − 1.01i)16-s + (−0.189 − 0.327i)17-s − 0.368i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.702 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67044 - 0.698202i\)
\(L(\frac12)\) \(\approx\) \(1.67044 - 0.698202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-1.35 - 0.780i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + (0.486 - 0.280i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.780 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.16 + 3.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.56iT - 31T^{2} \)
37 \( 1 + (-6.54 - 3.78i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.35 - 0.780i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 + (-2.49 + 1.43i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.95 + 2.28i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.1 + 7i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.1iT - 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 + (-4.22 - 2.43i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.41 - 4.28i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.91064860651528108141758116661, −9.609501744355988000374292480701, −9.226544121314750707447625130597, −8.085722571626584862333703540208, −6.88794792765682065548284937560, −5.69621139016220384193478187363, −5.08715592264018593163297223870, −4.52822030473387907445217988637, −3.24786289232962283447914017046, −0.857135967625351433750783888250, 2.16898152478762723623209492597, 3.08573410359696370047508131333, 4.02752064850149023721945472630, 5.46667144049464516100205027440, 6.21066395608825976567734021273, 7.39882917284483392281500600296, 7.898337664275891601073964492557, 9.574335770092639275324299555935, 10.50831547896663225809614951745, 11.25455769011499648581863344912

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