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Properties

Label 2-260-13.10-c1-0-0
Degree $2$
Conductor $260$
Sign $0.184 - 0.982i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
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  + (1.16 + 2.01i)3-s + i·5-s + (0.346 + 0.199i)7-s + (−1.21 + 2.11i)9-s + (1.5 − 0.866i)11-s + (−0.619 + 3.55i)13-s + (−2.01 + 1.16i)15-s + (0.346 − 0.599i)17-s + (−4.65 − 2.68i)19-s + 0.932i·21-s + (−0.0535 − 0.0927i)23-s − 25-s + 1.30·27-s + (2.45 + 4.24i)29-s − 7.86i·31-s + ⋯
L(s)  = 1  + (0.673 + 1.16i)3-s + 0.447i·5-s + (0.130 + 0.0755i)7-s + (−0.406 + 0.704i)9-s + (0.452 − 0.261i)11-s + (−0.171 + 0.985i)13-s + (−0.521 + 0.301i)15-s + (0.0839 − 0.145i)17-s + (−1.06 − 0.616i)19-s + 0.203i·21-s + (−0.0111 − 0.0193i)23-s − 0.200·25-s + 0.251·27-s + (0.455 + 0.788i)29-s − 1.41i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.184 - 0.982i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18858 + 0.986289i\)
\(L(\frac12)\) \(\approx\) \(1.18858 + 0.986289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + (0.619 - 3.55i)T \)
good3 \( 1 + (-1.16 - 2.01i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.346 - 0.199i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.346 + 0.599i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.65 + 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0535 + 0.0927i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.45 - 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 + 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (6.30 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.34 - 7.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.15 - 0.664i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.35 + 1.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 14.0iT - 83T^{2} \)
89 \( 1 + (-0.300 + 0.173i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.66 + 4.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.00345321273218051001807940088, −11.05111160812326143080648981663, −10.25404902024389407337015850700, −9.225978799433715620586240149124, −8.729521673169633921189220175395, −7.32544706471570847626461910225, −6.16111003309389440926264402037, −4.61980104263590401138682521249, −3.81196785478660970369013151617, −2.45996850235239659036784096039, 1.35036155603496843858478719400, 2.75674761174743290793420948402, 4.36805520518789901397622507788, 5.88054320077329175923600944085, 6.97019869928778279039730819895, 7.991602325944617921833052111157, 8.526298088903630000738259091662, 9.761417383672499537767625088498, 10.86608336862397759680008309267, 12.24903023573867635684644506839

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