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Properties

Label 2-130-65.49-c1-0-1
Degree $2$
Conductor $130$
Sign $0.496 - 0.867i$
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  + (−0.5 − 0.866i)2-s + (−1.01 + 0.588i)3-s + (−0.499 + 0.866i)4-s + (0.235 + 2.22i)5-s + (1.01 + 0.588i)6-s + (−1.04 + 1.80i)7-s + 0.999·8-s + (−0.807 + 1.39i)9-s + (1.80 − 1.31i)10-s + (2.59 − 1.49i)11-s − 1.17i·12-s + (1.51 + 3.27i)13-s + 2.08·14-s + (−1.54 − 2.12i)15-s + (−0.5 − 0.866i)16-s + (1.09 + 0.630i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.588 + 0.339i)3-s + (−0.249 + 0.433i)4-s + (0.105 + 0.994i)5-s + (0.415 + 0.240i)6-s + (−0.394 + 0.683i)7-s + 0.353·8-s + (−0.269 + 0.466i)9-s + (0.571 − 0.416i)10-s + (0.781 − 0.451i)11-s − 0.339i·12-s + (0.421 + 0.906i)13-s + 0.557·14-s + (−0.399 − 0.549i)15-s + (−0.125 − 0.216i)16-s + (0.264 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.496 - 0.867i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.496 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593950 + 0.344463i\)
\(L(\frac12)\) \(\approx\) \(0.593950 + 0.344463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.235 - 2.22i)T \)
13 \( 1 + (-1.51 - 3.27i)T \)
good3 \( 1 + (1.01 - 0.588i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.04 - 1.80i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 + 1.49i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.09 - 0.630i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.37 + 1.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.778 + 0.449i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.235 + 0.407i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.277iT - 31T^{2} \)
37 \( 1 + (-3.51 - 6.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-9.66 + 5.58i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.74 + 3.89i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 + (-10.7 - 6.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.82 - 6.61i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.07 - 2.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (5.98 - 3.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.28 - 2.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.45059995421395220924921023459, −12.04793213884760360898277948936, −11.30931509694044575553119157600, −10.60836947748897612532490124461, −9.492138803817946905723249718155, −8.464627401268105823642960151662, −6.81389360128665836378422333818, −5.78600773030713859183683066074, −4.01372597711788230737255708628, −2.46445432879887936497326973791, 0.924146857455589002477308090002, 4.06316575708088113164335987039, 5.57857892518856183264273593148, 6.48081030207991452741001066963, 7.70848556024502492726545070796, 8.898171965947646456644050528877, 9.818867335019769476798199220056, 11.06129090518071444633032253960, 12.35902559788257907237050163155, 12.97571584826819355130086990490

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