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Properties

Label 2-130-65.47-c1-0-2
Degree $2$
Conductor $130$
Sign $0.993 - 0.112i$
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 + (−1.15 + 1.15i)3-s − 4-s + (1.65 + 1.5i)5-s + (1.15 + 1.15i)6-s + 3·7-s + i·8-s + 0.316i·9-s + (1.5 − 1.65i)10-s + (3.31 − 3.31i)11-s + (1.15 − 1.15i)12-s + (−3 + 2i)13-s − 3i·14-s + (−3.65 + 0.183i)15-s + 16-s + (−3.15 + 3.15i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.668 + 0.668i)3-s − 0.5·4-s + (0.741 + 0.670i)5-s + (0.472 + 0.472i)6-s + 1.13·7-s + 0.353i·8-s + 0.105i·9-s + (0.474 − 0.524i)10-s + (1.00 − 1.00i)11-s + (0.334 − 0.334i)12-s + (−0.832 + 0.554i)13-s − 0.801i·14-s + (−0.944 + 0.0473i)15-s + 0.250·16-s + (−0.766 + 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.993 - 0.112i$
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: \((2,\ 130,\ (\ :1/2),\ 0.993 - 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01589 + 0.0570826i\)
\(L(\frac12)\) \(\approx\) \(1.01589 + 0.0570826i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 - 1.5i)T \)
13 \( 1 + (3 - 2i)T \)
good3 \( 1 + (1.15 - 1.15i)T - 3iT^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + (-3.31 + 3.31i)T - 11iT^{2} \)
17 \( 1 + (3.15 - 3.15i)T - 17iT^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 + (3.31 + 3.31i)T + 23iT^{2} \)
29 \( 1 + 6.31iT - 29T^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + (6.31 + 6.31i)T + 41iT^{2} \)
43 \( 1 + (-2.47 - 2.47i)T + 43iT^{2} \)
47 \( 1 + 9.31T + 47T^{2} \)
53 \( 1 + (-9.63 + 9.63i)T - 53iT^{2} \)
59 \( 1 + (-0.316 - 0.316i)T + 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4.94iT - 67T^{2} \)
71 \( 1 + (2.84 + 2.84i)T + 71iT^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 12.9iT - 79T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 + (-0.316 - 0.316i)T + 89iT^{2} \)
97 \( 1 - 8.94iT - 97T^{2} \)
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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

−13.46921300489278315036024924363, −11.77855932080962877575641752925, −11.27797645355773336438294468932, −10.45776097518357187206765390509, −9.507378864288517136660624141310, −8.261048674711215591901401423258, −6.49379453968045549775435707444, −5.25376304505927525895188092657, −4.11231289958349244744659309402, −2.12069995446381468213987645657, 1.52685583126304549845630754145, 4.61301069285376201831877606548, 5.51601220307747301594781447428, 6.72835072096288243517597233164, 7.68959152861224254696584351895, 9.024943728555946670271473252843, 9.943878323259637616898362751324, 11.63576426191260636346735977461, 12.27496266241979411528259645129, 13.29898795407723739480962772167

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