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Properties

Label 2-130-65.18-c1-0-0
Degree $2$
Conductor $130$
Sign $0.134 - 0.990i$
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 + (−1 − i)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 + (−0.242 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.134 - 0.990i$
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: \((2,\ 130,\ (\ :1/2),\ 0.134 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.954349 + 0.833303i\)
\(L(\frac12)\) \(\approx\) \(0.954349 + 0.833303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (7 - 7i)T - 41iT^{2} \)
43 \( 1 + (-1 + i)T - 43iT^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (9 - 9i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + (-5 + 5i)T - 71iT^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (-11 + 11i)T - 89iT^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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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

−13.64337110530095432106357131999, −13.05459281641719259178489730271, −11.35548037414773873533430140691, −9.988714052225930792873778562481, −9.396034766172495879402545659133, −8.436428541187349539410498002341, −6.74498304168338119841727422465, −6.08369063518413757126183913516, −4.36437902718636375213579142766, −2.88718171924783037467530224203, 1.78487513909741876656119990510, 3.17974475417872030817262495352, 5.07529002721091630239953100297, 6.42182849287379615300516084221, 8.002957984633436240618410672943, 8.925371997439647766362272555654, 10.03163407259311614154419274879, 10.82998032523111201228315194521, 12.59885894662536654305191701361, 12.92358644331471854043348906361

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