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Properties

Label 2-130-13.3-c1-0-1
Degree $2$
Conductor $130$
Sign $0.0128 - 0.999i$
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  + (0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 + 0.866i)4-s − 5-s + (−0.999 + 1.73i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−1.5 − 2.59i)11-s − 1.99·12-s + (2.5 + 2.59i)13-s + 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.408 + 0.707i)6-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.452 − 0.783i)11-s − 0.577·12-s + (0.693 + 0.720i)13-s + 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984484 + 0.971940i\)
\(L(\frac12)\) \(\approx\) \(0.984484 + 0.971940i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + (-2.5 - 2.59i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.12784457836087530816329026197, −12.71965370516653741025565381520, −11.47308005407024246172270969987, −10.39801871047967570866772633202, −9.179995336337289743011884801363, −8.326706737784315408192174693614, −7.15664739908629145125579030445, −5.60566696672177544042946923934, −4.25266517359728745125292576278, −3.35015108045084868786478926176, 1.78280070726498445058305761751, 3.27272476255967007787113062498, 4.95083955912040390605254314489, 6.52583122783701339232766716257, 7.85652067569092072475132031267, 8.585720756094299345352879469530, 10.14577725609900815994866656798, 11.14897435120650597554620038914, 12.43652159705278810541361715704, 12.85814860192904584263256400795

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