Properties

Label 2-260-65.2-c1-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 + 0.5i)3-s + (−1 − 2i)5-s + (3.23 + 1.86i)7-s + (0.633 + 0.366i)9-s + (−0.598 + 2.23i)11-s + (3 − 2i)13-s + (−0.866 − 4.23i)15-s + (−1.13 − 4.23i)17-s + (0.866 − 0.232i)19-s + (5.09 + 5.09i)21-s + (−1.86 + 6.96i)23-s + (−3 + 4i)25-s + (−3.09 − 3.09i)27-s + (−7.96 + 4.59i)29-s + (5.73 − 5.73i)31-s + ⋯
L(s)  = 1  + (1.07 + 0.288i)3-s + (−0.447 − 0.894i)5-s + (1.22 + 0.705i)7-s + (0.211 + 0.122i)9-s + (−0.180 + 0.672i)11-s + (0.832 − 0.554i)13-s + (−0.223 − 1.09i)15-s + (−0.275 − 1.02i)17-s + (0.198 − 0.0532i)19-s + (1.11 + 1.11i)21-s + (−0.389 + 1.45i)23-s + (−0.600 + 0.800i)25-s + (−0.596 − 0.596i)27-s + (−1.47 + 0.853i)29-s + (1.02 − 1.02i)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: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.999 - 0.00863i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74978 + 0.00755749i\)
\(L(\frac12)\) \(\approx\) \(1.74978 + 0.00755749i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
13 \( 1 + (-3 + 2i)T \)
good3 \( 1 + (-1.86 - 0.5i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-3.23 - 1.86i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.598 - 2.23i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.13 + 4.23i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.232i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.86 - 6.96i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.96 - 4.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.73 + 5.73i)T - 31iT^{2} \)
37 \( 1 + (-0.232 + 0.133i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.133 + 0.0358i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (11.3 - 3.03i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.535iT - 47T^{2} \)
53 \( 1 + (1.53 - 1.53i)T - 53iT^{2} \)
59 \( 1 + (1.79 + 6.69i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 + 4.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.13 - 4.23i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 4.53iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-14.7 - 3.96i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.23 - 3.86i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.80814096893286907262940895879, −11.33917978587305487416166604916, −9.704307183366894022667824049628, −9.000065615346826524246752567429, −8.177037954657857910241439387424, −7.59310500925393735859461081877, −5.58570420194405543104306933732, −4.67307418932288614448509143386, −3.39786197162718606891303099303, −1.83275537085266612089379124998, 1.90331966258010760259918966033, 3.34057032696346975032708691445, 4.34664307617958722339205963536, 6.15443319531702183703291868572, 7.31063719442197265825231105331, 8.230612765454673446587582806981, 8.610346158726420824751441664746, 10.30522598646174321086962799867, 11.00884395369784834908030458418, 11.76254118221593926638461700712

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