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Properties

Label 2-130-13.4-c1-0-2
Degree $2$
Conductor $130$
Sign $0.964 + 0.265i$
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  + (−0.866 − 0.5i)2-s + (−0.366 + 0.633i)3-s + (0.499 + 0.866i)4-s − i·5-s + (0.633 − 0.366i)6-s + (2.59 − 1.5i)7-s − 0.999i·8-s + (1.23 + 2.13i)9-s + (−0.5 + 0.866i)10-s + (2.59 + 1.5i)11-s − 0.732·12-s + (3.5 − 0.866i)13-s − 3·14-s + (0.633 + 0.366i)15-s + (−0.5 + 0.866i)16-s + (−4.09 − 7.09i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.211 + 0.366i)3-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (0.258 − 0.149i)6-s + (0.981 − 0.566i)7-s − 0.353i·8-s + (0.410 + 0.711i)9-s + (−0.158 + 0.273i)10-s + (0.783 + 0.452i)11-s − 0.211·12-s + (0.970 − 0.240i)13-s − 0.801·14-s + (0.163 + 0.0945i)15-s + (−0.125 + 0.216i)16-s + (−0.993 − 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866281 - 0.116874i\)
\(L(\frac12)\) \(\approx\) \(0.866281 - 0.116874i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + iT \)
13 \( 1 + (-3.5 + 0.866i)T \)
good3 \( 1 + (0.366 - 0.633i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.09 + 7.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.401 - 0.232i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.73 - 8.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73iT - 31T^{2} \)
37 \( 1 + (0.696 + 0.401i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9 + 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (-9 + 5.19i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.19 - 3i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 4.19T + 79T^{2} \)
83 \( 1 + 8.19iT - 83T^{2} \)
89 \( 1 + (-5.89 - 3.40i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.90 + 4.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.35431885580636845362666310962, −11.81272724583241790205629408988, −11.24143945034493389253295960369, −10.18823552287621849241617557095, −9.177437026845674059168669774180, −8.052891047627317324138043260733, −7.02761420502294060023201411231, −5.13314679161140245856225262785, −4.01592167938776934834061317335, −1.64170616099128021979535656547, 1.73049891669617410671528272357, 4.11243522165841246154540440590, 6.10361021065949761062404583891, 6.61392585311012895962959764281, 8.255465819569279081369416427980, 8.811301885522161806246933870043, 10.32401655821139050957277387053, 11.26924998963423644375373818350, 12.10270905431703890984230128258, 13.41087801307522123354666433413

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