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Properties

Label 2-507-39.2-c1-0-10
Degree $2$
Conductor $507$
Sign $-0.964 + 0.262i$
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  + (−0.522 + 1.94i)2-s + (0.152 + 1.72i)3-s + (−1.79 − 1.03i)4-s + (1.72 + 1.72i)5-s + (−3.44 − 0.603i)6-s + (3.00 − 0.805i)7-s + (0.105 − 0.105i)8-s + (−2.95 + 0.526i)9-s + (−4.25 + 2.45i)10-s + (2.67 + 0.716i)11-s + (1.51 − 3.25i)12-s + 6.28i·14-s + (−2.70 + 3.23i)15-s + (−1.92 − 3.33i)16-s + (−2.89 + 5.01i)17-s + (0.515 − 6.03i)18-s + ⋯
L(s)  = 1  + (−0.369 + 1.37i)2-s + (0.0881 + 0.996i)3-s + (−0.898 − 0.518i)4-s + (0.770 + 0.770i)5-s + (−1.40 − 0.246i)6-s + (1.13 − 0.304i)7-s + (0.0372 − 0.0372i)8-s + (−0.984 + 0.175i)9-s + (−1.34 + 0.777i)10-s + (0.806 + 0.215i)11-s + (0.437 − 0.940i)12-s + 1.67i·14-s + (−0.699 + 0.835i)15-s + (−0.480 − 0.832i)16-s + (−0.701 + 1.21i)17-s + (0.121 − 1.42i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.964 + 0.262i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192091 - 1.43508i\)
\(L(\frac12)\) \(\approx\) \(0.192091 - 1.43508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.152 - 1.72i)T \)
13 \( 1 \)
good2 \( 1 + (0.522 - 1.94i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.72 - 1.72i)T + 5iT^{2} \)
7 \( 1 + (-3.00 + 0.805i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-2.67 - 0.716i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.89 - 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.388 - 1.44i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.93 + 3.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.53 + 1.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.56 + 3.56i)T - 31iT^{2} \)
37 \( 1 + (1.03 - 3.87i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.75 + 6.55i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.59 - 0.921i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.115 + 0.115i)T - 47iT^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 + (0.446 + 1.66i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.69 + 4.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.1 + 3.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.326 + 0.0875i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.54 - 8.54i)T + 73iT^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + (-2.83 - 2.83i)T + 83iT^{2} \)
89 \( 1 + (4.10 + 1.10i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.56 - 9.55i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.05582325977618677180132456221, −10.32380107770671238946196801423, −9.495252087042635013388874967997, −8.514265575017083642269992212361, −7.962587050356438767168870224450, −6.62401339307957164876998668410, −6.09648160345398176803995188022, −4.98176414841940005349990237999, −4.01821832716876090748125415762, −2.26333720360779428814496024474, 1.06327813267656866260841232886, 1.83016880019118245606031723088, 2.88873453840085707338048594373, 4.55838853005171702735759958290, 5.66469522276184864970194601976, 6.80442180542747969677904075652, 8.052784276881886734931612119689, 9.078993999623509082572001650657, 9.244437868017961000059853035027, 10.66155717187543817460943115114

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