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Properties

Label 2-130-65.37-c1-0-2
Degree $2$
Conductor $130$
Sign $0.612 - 0.790i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
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  + (−0.866 + 0.5i)2-s + (2.38 + 0.640i)3-s + (0.499 − 0.866i)4-s + (−0.606 + 2.15i)5-s + (−2.38 + 0.640i)6-s + (0.551 − 0.954i)7-s + 0.999i·8-s + (2.70 + 1.56i)9-s + (−0.551 − 2.16i)10-s + (−3.23 − 0.865i)11-s + (1.74 − 1.74i)12-s + (1.57 − 3.24i)13-s + 1.10i·14-s + (−2.82 + 4.75i)15-s + (−0.5 − 0.866i)16-s + (0.749 + 2.79i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (1.37 + 0.369i)3-s + (0.249 − 0.433i)4-s + (−0.271 + 0.962i)5-s + (−0.975 + 0.261i)6-s + (0.208 − 0.360i)7-s + 0.353i·8-s + (0.901 + 0.520i)9-s + (−0.174 − 0.685i)10-s + (−0.974 − 0.261i)11-s + (0.505 − 0.505i)12-s + (0.437 − 0.899i)13-s + 0.294i·14-s + (−0.730 + 1.22i)15-s + (−0.125 − 0.216i)16-s + (0.181 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03710 + 0.508587i\)
\(L(\frac12)\) \(\approx\) \(1.03710 + 0.508587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.606 - 2.15i)T \)
13 \( 1 + (-1.57 + 3.24i)T \)
good3 \( 1 + (-2.38 - 0.640i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.551 + 0.954i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.23 + 0.865i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.749 - 2.79i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.15 - 4.29i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.93 + 7.23i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.66 - 0.963i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.17 + 3.17i)T + 31iT^{2} \)
37 \( 1 + (5.46 + 9.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.348 - 1.30i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.64 + 1.24i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 7.88T + 47T^{2} \)
53 \( 1 + (8.09 - 8.09i)T - 53iT^{2} \)
59 \( 1 + (9.63 - 2.58i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.68 - 6.38i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.69 + 2.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.81 + 1.28i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 16.4iT - 73T^{2} \)
79 \( 1 + 0.747iT - 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-0.953 + 3.56i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.98 - 2.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.92480755811772154086737165205, −12.62834718604887378135054887919, −10.65172272550569682797938435832, −10.52857099968117749157103673235, −9.093674451060304686500356027618, −8.021809415169585717577758457239, −7.53976467964618737780988495026, −5.88822552857850092252391696777, −3.81518423070482049123399368079, −2.59148102713572964137404197343, 1.82595513268569819686332505645, 3.30317011769006619153082888243, 5.02842609464618127071047468969, 7.23615320027313819603906059620, 8.057923415909692241374669608046, 8.996074107799511449965779789998, 9.500440930442475697335436829492, 11.18616945900383541300848766462, 12.19867825179463261444191264181, 13.27857449124070854135440740143

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