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Properties

Label 2-260-20.3-c1-0-24
Degree $2$
Conductor $260$
Sign $-0.0357 + 0.999i$
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  + (−1.26 − 0.629i)2-s + (2.21 − 2.21i)3-s + (1.20 + 1.59i)4-s + (1.36 + 1.77i)5-s + (−4.19 + 1.40i)6-s + (−2.47 − 2.47i)7-s + (−0.523 − 2.77i)8-s − 6.78i·9-s + (−0.609 − 3.10i)10-s + 1.32i·11-s + (6.19 + 0.859i)12-s + (0.707 + 0.707i)13-s + (1.57 + 4.70i)14-s + (6.93 + 0.906i)15-s + (−1.08 + 3.84i)16-s + (4.38 − 4.38i)17-s + ⋯
L(s)  = 1  + (−0.895 − 0.445i)2-s + (1.27 − 1.27i)3-s + (0.603 + 0.797i)4-s + (0.609 + 0.792i)5-s + (−1.71 + 0.574i)6-s + (−0.937 − 0.937i)7-s + (−0.184 − 0.982i)8-s − 2.26i·9-s + (−0.192 − 0.981i)10-s + 0.400i·11-s + (1.78 + 0.248i)12-s + (0.196 + 0.196i)13-s + (0.421 + 1.25i)14-s + (1.79 + 0.233i)15-s + (−0.272 + 0.962i)16-s + (1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0357 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0357 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.0357 + 0.999i$
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: \((2,\ 260,\ (\ :1/2),\ -0.0357 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886606 - 0.918935i\)
\(L(\frac12)\) \(\approx\) \(0.886606 - 0.918935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.629i)T \)
5 \( 1 + (-1.36 - 1.77i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-2.21 + 2.21i)T - 3iT^{2} \)
7 \( 1 + (2.47 + 2.47i)T + 7iT^{2} \)
11 \( 1 - 1.32iT - 11T^{2} \)
17 \( 1 + (-4.38 + 4.38i)T - 17iT^{2} \)
19 \( 1 - 0.135T + 19T^{2} \)
23 \( 1 + (-0.621 + 0.621i)T - 23iT^{2} \)
29 \( 1 - 2.54iT - 29T^{2} \)
31 \( 1 - 3.33iT - 31T^{2} \)
37 \( 1 + (6.56 - 6.56i)T - 37iT^{2} \)
41 \( 1 - 3.69T + 41T^{2} \)
43 \( 1 + (4.48 - 4.48i)T - 43iT^{2} \)
47 \( 1 + (-0.681 - 0.681i)T + 47iT^{2} \)
53 \( 1 + (-7.99 - 7.99i)T + 53iT^{2} \)
59 \( 1 + 3.78T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + (-5.75 - 5.75i)T + 67iT^{2} \)
71 \( 1 - 4.51iT - 71T^{2} \)
73 \( 1 + (3.49 + 3.49i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (-2.19 + 2.19i)T - 83iT^{2} \)
89 \( 1 + 5.56iT - 89T^{2} \)
97 \( 1 + (0.789 - 0.789i)T - 97iT^{2} \)
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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

−11.87617980951748391497876395475, −10.46484425612886925705134039298, −9.722039896219991424090492886877, −8.957504792049196930032572548339, −7.68088969950617927683735881160, −7.09835315864216411852405985593, −6.46035288899837370127543322913, −3.44425574100712974378391805693, −2.77993357181372245727250221792, −1.34553847456218880526884101523, 2.21287299971878143031608640098, 3.54666732654761986624211592474, 5.25345604614524252057990794622, 6.05849113422959355219090840786, 7.903965815685559779259730696745, 8.687147093261450787448160246006, 9.247031111698902644114745908761, 9.918141772050759552533970304877, 10.64422817743420676515339138991, 12.23361711089735437693998919651

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