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Properties

Label 2-130-65.18-c1-0-2
Degree $2$
Conductor $130$
Sign $0.0120 - 0.999i$
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  + i·2-s + (2.15 + 2.15i)3-s − 4-s + (−1.65 − 1.5i)5-s + (−2.15 + 2.15i)6-s + 3·7-s − i·8-s + 6.31i·9-s + (1.5 − 1.65i)10-s + (−3.31 − 3.31i)11-s + (−2.15 − 2.15i)12-s + (−3 − 2i)13-s + 3i·14-s + (−0.341 − 6.81i)15-s + 16-s + (0.158 + 0.158i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (1.24 + 1.24i)3-s − 0.5·4-s + (−0.741 − 0.670i)5-s + (−0.881 + 0.881i)6-s + 1.13·7-s − 0.353i·8-s + 2.10i·9-s + (0.474 − 0.524i)10-s + (−1.00 − 1.00i)11-s + (−0.623 − 0.623i)12-s + (−0.832 − 0.554i)13-s + 0.801i·14-s + (−0.0882 − 1.76i)15-s + 0.250·16-s + (0.0383 + 0.0383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0120 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0120 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.0120 - 0.999i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.0120 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.968265 + 0.956626i\)
\(L(\frac12)\) \(\approx\) \(0.968265 + 0.956626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.65 + 1.5i)T \)
13 \( 1 + (3 + 2i)T \)
good3 \( 1 + (-2.15 - 2.15i)T + 3iT^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + (3.31 + 3.31i)T + 11iT^{2} \)
17 \( 1 + (-0.158 - 0.158i)T + 17iT^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 + (-3.31 + 3.31i)T - 23iT^{2} \)
29 \( 1 + 0.316iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (-0.316 + 0.316i)T - 41iT^{2} \)
43 \( 1 + (7.47 - 7.47i)T - 43iT^{2} \)
47 \( 1 + 2.68T + 47T^{2} \)
53 \( 1 + (3.63 + 3.63i)T + 53iT^{2} \)
59 \( 1 + (6.31 - 6.31i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 14.9iT - 67T^{2} \)
71 \( 1 + (6.15 - 6.15i)T - 71iT^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 6.94iT - 79T^{2} \)
83 \( 1 + 0.316T + 83T^{2} \)
89 \( 1 + (6.31 - 6.31i)T - 89iT^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−13.94493867144122664580674533075, −12.89147493056664678421587539922, −11.30469688267944653636104376165, −10.23414018000228375172576089732, −9.066063303693335511715761406309, −8.058697020141944729999862625093, −7.898377072013147318414054489883, −5.22596456567608498812029668333, −4.57714053866190848076928242752, −3.13664288048129297155868845406, 1.93313057832796320205704115891, 3.02823978833235987402365761481, 4.72096310626181716814485448599, 7.15597017497016994118066883267, 7.63467490674638925596870736527, 8.622471267111808611510759585970, 9.910454854822502961071666982127, 11.31073112517267043946255907059, 12.09758039163621013493933598390, 13.02218695817276505534377080536

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