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Properties

Label 2-130-65.49-c1-0-6
Degree $2$
Conductor $130$
Sign $0.0510 + 0.998i$
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 + (1.59 − 0.918i)3-s + (−0.499 + 0.866i)4-s + (−2.21 − 0.339i)5-s + (−1.59 − 0.918i)6-s + (2.39 − 4.15i)7-s + 0.999·8-s + (0.188 − 0.326i)9-s + (0.811 + 2.08i)10-s + (1.43 − 0.825i)11-s + 1.83i·12-s + (−1.09 + 3.43i)13-s − 4.79·14-s + (−3.82 + 1.49i)15-s + (−0.5 − 0.866i)16-s + (−0.0697 − 0.0402i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.918 − 0.530i)3-s + (−0.249 + 0.433i)4-s + (−0.988 − 0.151i)5-s + (−0.649 − 0.375i)6-s + (0.906 − 1.57i)7-s + 0.353·8-s + (0.0628 − 0.108i)9-s + (0.256 + 0.658i)10-s + (0.431 − 0.248i)11-s + 0.530i·12-s + (−0.302 + 0.953i)13-s − 1.28·14-s + (−0.988 + 0.385i)15-s + (−0.125 − 0.216i)16-s + (−0.0169 − 0.00976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0510 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0510 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.0510 + 0.998i$
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: \((2,\ 130,\ (\ :1/2),\ 0.0510 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780593 - 0.741742i\)
\(L(\frac12)\) \(\approx\) \(0.780593 - 0.741742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 0.339i)T \)
13 \( 1 + (1.09 - 3.43i)T \)
good3 \( 1 + (-1.59 + 0.918i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.39 + 4.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.43 + 0.825i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.0697 + 0.0402i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.67 - 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.10 - 2.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.21 - 3.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.06iT - 31T^{2} \)
37 \( 1 + (-0.908 - 1.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.87 - 3.39i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.35 - 4.24i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.448T + 47T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (-1.82 - 1.05i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.36 + 3.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + 12.4T + 83T^{2} \)
89 \( 1 + (-7.23 + 4.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.90 - 5.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.16380537466733784899335262390, −11.79591286517597404878697574949, −11.22831120367608054136424217575, −9.959969997996276158526022390449, −8.624098698609899895224253423697, −7.78820650974230872795388987426, −7.19164653898845313469651101414, −4.51487911498066216539448404581, −3.50446013331580491312432713442, −1.49140672827941370793517788811, 2.76650445435974183825731678520, 4.41599723686787318792423401473, 5.74870823462508683364763830396, 7.47736188492079133731431248631, 8.434439659421984342556333168026, 8.931151934517429863234172356399, 10.18553623292000288945662947539, 11.62550961522850466391746166168, 12.33604125235756609204577378023, 14.14689436620066960762427140859

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