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Properties

Label 2-260-65.2-c1-0-6
Degree $2$
Conductor $260$
Sign $-0.999 + 0.00863i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.133i)3-s + (−2 + i)5-s + (−2.13 − 1.23i)7-s + (−2.36 − 1.36i)9-s + (−1.13 + 4.23i)11-s + (−2 − 3i)13-s + (1.13 − 0.232i)15-s + (−0.232 − 0.866i)17-s + (−2.86 + 0.767i)19-s + (0.901 + 0.901i)21-s + (−0.0358 + 0.133i)23-s + (3 − 4i)25-s + (2.09 + 2.09i)27-s + (−1.5 + 0.866i)29-s + (−5.19 + 5.19i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.0773i)3-s + (−0.894 + 0.447i)5-s + (−0.806 − 0.465i)7-s + (−0.788 − 0.455i)9-s + (−0.341 + 1.27i)11-s + (−0.554 − 0.832i)13-s + (0.292 − 0.0599i)15-s + (−0.0562 − 0.210i)17-s + (−0.657 + 0.176i)19-s + (0.196 + 0.196i)21-s + (−0.00748 + 0.0279i)23-s + (0.600 − 0.800i)25-s + (0.403 + 0.403i)27-s + (−0.278 + 0.160i)29-s + (−0.933 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.999 + 0.00863i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.999 + 0.00863i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
13 \( 1 + (2 + 3i)T \)
good3 \( 1 + (0.5 + 0.133i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.13 + 1.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.13 - 4.23i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.232 + 0.866i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.86 - 0.767i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.0358 - 0.133i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.19 - 5.19i)T - 31iT^{2} \)
37 \( 1 + (-1.33 + 0.767i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.33 - 2.5i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.96 - 1.59i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 + (-2.46 + 2.46i)T - 53iT^{2} \)
59 \( 1 + (-2.33 - 8.69i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.13 - 10.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.598 - 2.23i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 - 0.535iT - 79T^{2} \)
83 \( 1 + 2.92iT - 83T^{2} \)
89 \( 1 + (14.7 + 3.96i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.86 + 6.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.56704897213737104441459579286, −10.55781865812136319524366306394, −9.814470120017475705842325128333, −8.515298786171705170278162602288, −7.36505334215355171349089517679, −6.72843737924971687083652450378, −5.34669825879048763921914900698, −3.97357922874094054930924309737, −2.80424179502574274612603317615, 0, 2.73706155896064975018308445461, 4.08168167632930616719058901640, 5.40279072283639638831662990651, 6.33284359158995376358876461607, 7.71520440852332289957240188249, 8.631448819359183117117685580304, 9.413793070350809790817696357432, 10.91016409936881307445572337498, 11.40930939569602749164264107440

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