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Properties

Label 2-130-65.64-c1-0-4
Degree $2$
Conductor $130$
Sign $0.829 - 0.558i$
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  + 2-s + 0.792i·3-s + 4-s + (0.686 + 2.12i)5-s + 0.792i·6-s − 3.37·7-s + 8-s + 2.37·9-s + (0.686 + 2.12i)10-s − 5.04i·11-s + 0.792i·12-s + (−1 − 3.46i)13-s − 3.37·14-s + (−1.68 + 0.543i)15-s + 16-s + 2.67i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.457i·3-s + 0.5·4-s + (0.306 + 0.951i)5-s + 0.323i·6-s − 1.27·7-s + 0.353·8-s + 0.790·9-s + (0.216 + 0.672i)10-s − 1.52i·11-s + 0.228i·12-s + (−0.277 − 0.960i)13-s − 0.901·14-s + (−0.435 + 0.140i)15-s + 0.250·16-s + 0.648i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.829 - 0.558i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45816 + 0.445411i\)
\(L(\frac12)\) \(\approx\) \(1.45816 + 0.445411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-0.686 - 2.12i)T \)
13 \( 1 + (1 + 3.46i)T \)
good3 \( 1 - 0.792iT - 3T^{2} \)
7 \( 1 + 3.37T + 7T^{2} \)
11 \( 1 + 5.04iT - 11T^{2} \)
17 \( 1 - 2.67iT - 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 6.63iT - 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 8.11T + 37T^{2} \)
41 \( 1 - 3.16iT - 41T^{2} \)
43 \( 1 - 9.30iT - 43T^{2} \)
47 \( 1 + 4.62T + 47T^{2} \)
53 \( 1 - 1.58iT - 53T^{2} \)
59 \( 1 - 6.63iT - 59T^{2} \)
61 \( 1 - 4.74T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 3.96iT - 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 6.74T + 79T^{2} \)
83 \( 1 + 2.74T + 83T^{2} \)
89 \( 1 + 8.51iT - 89T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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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

−13.20331028439552013982926519297, −12.81215994893343790577697809048, −11.20741741914383710621280888568, −10.39683222419150761665280561550, −9.613316233769994500094521631356, −7.86910398140472636509266316202, −6.46679172930555845296954719401, −5.79755749479936508868821600553, −3.86552241175809994551681278818, −2.93727264539948136922182633616, 1.99227579991426789372538169091, 4.04603502305392209467504671752, 5.21183013172082247867660300186, 6.72225554438298111027548220518, 7.38190819636764343043864687498, 9.405304236531530087872051748019, 9.765754995382332918477746255798, 11.62230446068851039495357508573, 12.60065251576491717451602676939, 13.00811244730157629221031582301

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