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Properties

Label 2-260-260.207-c1-0-19
Degree $2$
Conductor $260$
Sign $0.990 - 0.136i$
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  + (1.40 − 0.178i)2-s + (1.93 − 0.5i)4-s + (−1.58 + 1.58i)5-s + (2.44 + 2.44i)7-s + (2.62 − 1.04i)8-s − 3i·9-s + (−1.93 + 2.50i)10-s + 2.44·11-s + (−3.58 + 0.418i)13-s + (3.87 + 3i)14-s + (3.50 − 1.93i)16-s + (−1 − i)17-s + (−0.534 − 4.20i)18-s + 2.44i·19-s + (−2.27 + 3.85i)20-s + ⋯
L(s)  = 1  + (0.992 − 0.126i)2-s + (0.968 − 0.250i)4-s + (−0.707 + 0.707i)5-s + (0.925 + 0.925i)7-s + (0.929 − 0.370i)8-s − i·9-s + (−0.612 + 0.790i)10-s + 0.738·11-s + (−0.993 + 0.116i)13-s + (1.03 + 0.801i)14-s + (0.875 − 0.484i)16-s + (−0.242 − 0.242i)17-s + (−0.126 − 0.992i)18-s + 0.561i·19-s + (−0.507 + 0.861i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.990 - 0.136i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17271 + 0.149381i\)
\(L(\frac12)\) \(\approx\) \(2.17271 + 0.149381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.178i)T \)
5 \( 1 + (1.58 - 1.58i)T \)
13 \( 1 + (3.58 - 0.418i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (3.87 + 3.87i)T + 23iT^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 9.79T + 31T^{2} \)
37 \( 1 + (6.32 - 6.32i)T - 37iT^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-7.74 - 7.74i)T + 43iT^{2} \)
47 \( 1 + (-2.44 - 2.44i)T + 47iT^{2} \)
53 \( 1 + (-2 + 2i)T - 53iT^{2} \)
59 \( 1 + 7.34iT - 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + (3.16 + 3.16i)T + 73iT^{2} \)
79 \( 1 - 7.74T + 79T^{2} \)
83 \( 1 + (4.89 - 4.89i)T - 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (-3.16 + 3.16i)T - 97iT^{2} \)
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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

−11.98616565638128373173062781775, −11.50695380710408181161718979915, −10.42722149384266867551327532931, −9.154772168031197021499304654887, −7.85896633181698049239704397348, −6.85976778422364253395213690754, −5.89964452628912234484363677981, −4.60151023960271146503900323452, −3.54996255205624054556078189226, −2.17551574265596023016345934681, 1.82956780279165619432200251846, 3.82791747069429316225903917768, 4.63519344470472726779828732929, 5.48444270566768739103864837141, 7.37475167135721831748976048818, 7.53460308677694883128715753868, 8.871248833655498188772265203240, 10.50466946030666662694610626674, 11.22411496683942233254937507844, 12.03170630442479971316593243145

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