Properties

Label 2-260-20.3-c1-0-23
Degree $2$
Conductor $260$
Sign $0.970 - 0.242i$
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  + (1.18 + 0.769i)2-s + (0.808 − 0.808i)3-s + (0.817 + 1.82i)4-s + (0.885 − 2.05i)5-s + (1.58 − 0.337i)6-s + (−0.771 − 0.771i)7-s + (−0.433 + 2.79i)8-s + 1.69i·9-s + (2.63 − 1.75i)10-s − 0.875i·11-s + (2.13 + 0.815i)12-s + (0.707 + 0.707i)13-s + (−0.322 − 1.50i)14-s + (−0.943 − 2.37i)15-s + (−2.66 + 2.98i)16-s + (4.91 − 4.91i)17-s + ⋯
L(s)  = 1  + (0.839 + 0.543i)2-s + (0.466 − 0.466i)3-s + (0.408 + 0.912i)4-s + (0.396 − 0.918i)5-s + (0.645 − 0.137i)6-s + (−0.291 − 0.291i)7-s + (−0.153 + 0.988i)8-s + 0.563i·9-s + (0.831 − 0.555i)10-s − 0.264i·11-s + (0.616 + 0.235i)12-s + (0.196 + 0.196i)13-s + (−0.0861 − 0.403i)14-s + (−0.243 − 0.613i)15-s + (−0.666 + 0.745i)16-s + (1.19 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24734 + 0.277107i\)
\(L(\frac12)\) \(\approx\) \(2.24734 + 0.277107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 - 0.769i)T \)
5 \( 1 + (-0.885 + 2.05i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.808 + 0.808i)T - 3iT^{2} \)
7 \( 1 + (0.771 + 0.771i)T + 7iT^{2} \)
11 \( 1 + 0.875iT - 11T^{2} \)
17 \( 1 + (-4.91 + 4.91i)T - 17iT^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
23 \( 1 + (6.03 - 6.03i)T - 23iT^{2} \)
29 \( 1 - 7.85iT - 29T^{2} \)
31 \( 1 + 5.45iT - 31T^{2} \)
37 \( 1 + (2.21 - 2.21i)T - 37iT^{2} \)
41 \( 1 + 1.36T + 41T^{2} \)
43 \( 1 + (-4.34 + 4.34i)T - 43iT^{2} \)
47 \( 1 + (3.81 + 3.81i)T + 47iT^{2} \)
53 \( 1 + (2.18 + 2.18i)T + 53iT^{2} \)
59 \( 1 - 0.749T + 59T^{2} \)
61 \( 1 + 4.79T + 61T^{2} \)
67 \( 1 + (-10.0 - 10.0i)T + 67iT^{2} \)
71 \( 1 + 5.01iT - 71T^{2} \)
73 \( 1 + (-6.85 - 6.85i)T + 73iT^{2} \)
79 \( 1 - 1.62T + 79T^{2} \)
83 \( 1 + (-2.66 + 2.66i)T - 83iT^{2} \)
89 \( 1 + 3.68iT - 89T^{2} \)
97 \( 1 + (-6.65 + 6.65i)T - 97iT^{2} \)
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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

−12.41262290844507645490067670553, −11.42113767022272828885965037390, −10.04760734136017112122979259608, −8.787372981915081224032946118891, −8.007936615773673317645027481353, −7.08121293514330968461404484841, −5.83712164118923681609555293492, −4.92423409291814848944065157349, −3.59826307756106261764956238888, −2.03130857408272735598983214035, 2.20747527047709293040728443710, 3.36970886045077268508952707211, 4.28444902502286175096684930488, 6.04658483699863496463866380833, 6.41786745411541645570402934453, 8.128202454691245312517302596348, 9.481718036920021204731180541685, 10.24350756475409623461790041293, 10.81474359291858662333049637464, 12.21749183032550536513395707636

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