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Properties

Label 2-130-65.47-c1-0-0
Degree $2$
Conductor $130$
Sign $-0.966 - 0.256i$
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 + (−1 + i)3-s − 4-s + (−2 + i)5-s + (−1 − i)6-s − 2·7-s − i·8-s + i·9-s + (−1 − 2i)10-s + (1 − i)11-s + (1 − i)12-s + (2 + 3i)13-s − 2i·14-s + (1 − 3i)15-s + 16-s + (−5 + 5i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.577 + 0.577i)3-s − 0.5·4-s + (−0.894 + 0.447i)5-s + (−0.408 − 0.408i)6-s − 0.755·7-s − 0.353i·8-s + 0.333i·9-s + (−0.316 − 0.632i)10-s + (0.301 − 0.301i)11-s + (0.288 − 0.288i)12-s + (0.554 + 0.832i)13-s − 0.534i·14-s + (0.258 − 0.774i)15-s + 0.250·16-s + (−1.21 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0752304 + 0.576389i\)
\(L(\frac12)\) \(\approx\) \(0.0752304 + 0.576389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
13 \( 1 + (-2 - 3i)T \)
good3 \( 1 + (1 - i)T - 3iT^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
17 \( 1 + (5 - 5i)T - 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (-5 - 5i)T + 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (1 - i)T - 53iT^{2} \)
59 \( 1 + (3 + 3i)T + 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + (-1 - i)T + 71iT^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-7 - 7i)T + 89iT^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−13.83482807008856876401964301951, −12.91384774346561695213202458657, −11.42861401356901615576254910237, −10.90128825868159979297770618239, −9.532483204309276384049169495982, −8.462245857808688188881549431520, −7.09628536207328513682712674365, −6.21776732484840220035337331889, −4.73357781054348967340208224176, −3.58082931103099909234163816111, 0.67393525040689026737603627805, 3.19134496946121484691862511641, 4.62825291288868211730395906024, 6.20244241655904005890301023883, 7.39296131743449674683871095964, 8.749487863837133232202493954313, 9.760400849918352357712739667087, 11.19680127344307284204668220015, 11.79624668082703837875315364206, 12.77399507881487237745451711120

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