Properties

Label 2-260-260.23-c1-0-35
Degree $2$
Conductor $260$
Sign $-0.884 + 0.466i$
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  + (0.366 − 1.36i)2-s + (−1.73 − i)4-s + (0.133 − 2.23i)5-s + (−2 + 1.99i)8-s + (−2.59 − 1.5i)9-s + (−2.99 − i)10-s + (1.59 − 3.23i)13-s + (1.99 + 3.46i)16-s + (0.0358 + 0.133i)17-s + (−3 + 3i)18-s + (−2.46 + 3.73i)20-s + (−4.96 − 0.598i)25-s + (−3.83 − 3.36i)26-s + (9.23 − 5.33i)29-s + (5.46 − 1.46i)32-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.0599 − 0.998i)5-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.948 − 0.316i)10-s + (0.443 − 0.896i)13-s + (0.499 + 0.866i)16-s + (0.00870 + 0.0324i)17-s + (−0.707 + 0.707i)18-s + (−0.550 + 0.834i)20-s + (−0.992 − 0.119i)25-s + (−0.751 − 0.660i)26-s + (1.71 − 0.989i)29-s + (0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273802 - 1.10626i\)
\(L(\frac12)\) \(\approx\) \(0.273802 - 1.10626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (-0.133 + 2.23i)T \)
13 \( 1 + (-1.59 + 3.23i)T \)
good3 \( 1 + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.0358 - 0.133i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-9.23 + 5.33i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (2.86 - 10.6i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-10.3 + 5.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5.29 - 5.29i)T + 53iT^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 2.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.16 - 1.16i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (17.7 - 4.75i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.84033389376177807957909186058, −10.71031258275367892220772446309, −9.772843918760856572033721003635, −8.772771927299445725384838735213, −8.166095409120280519768180337559, −6.12740126535897525430798517549, −5.25933939926732076545641023941, −4.07024148128909104046173651961, −2.74673051334977702377817061566, −0.866992351343229496059516783790, 2.78129623757159189916483365992, 4.12372830199504256312742633006, 5.50659100733429146511757047446, 6.44201946370872960518714546572, 7.30584786406429692685882214829, 8.371000029517526354214094418073, 9.272910550970650109539882760694, 10.53145971876565878089942232029, 11.43204248981908133472460572684, 12.50085451392675585442148941756

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