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Properties

Label 2-260-20.7-c1-0-22
Degree $2$
Conductor $260$
Sign $-0.120 + 0.992i$
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  + (0.203 − 1.39i)2-s + (−0.0775 − 0.0775i)3-s + (−1.91 − 0.569i)4-s + (0.871 + 2.05i)5-s + (−0.124 + 0.0927i)6-s + (2.89 − 2.89i)7-s + (−1.18 + 2.56i)8-s − 2.98i·9-s + (3.05 − 0.800i)10-s − 3.20i·11-s + (0.104 + 0.192i)12-s + (−0.707 + 0.707i)13-s + (−3.46 − 4.64i)14-s + (0.0921 − 0.227i)15-s + (3.35 + 2.18i)16-s + (−0.217 − 0.217i)17-s + ⋯
L(s)  = 1  + (0.143 − 0.989i)2-s + (−0.0447 − 0.0447i)3-s + (−0.958 − 0.284i)4-s + (0.389 + 0.920i)5-s + (−0.0507 + 0.0378i)6-s + (1.09 − 1.09i)7-s + (−0.419 + 0.907i)8-s − 0.995i·9-s + (0.967 − 0.253i)10-s − 0.965i·11-s + (0.0301 + 0.0556i)12-s + (−0.196 + 0.196i)13-s + (−0.925 − 1.24i)14-s + (0.0237 − 0.0586i)15-s + (0.837 + 0.545i)16-s + (−0.0528 − 0.0528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.909565 - 1.02620i\)
\(L(\frac12)\) \(\approx\) \(0.909565 - 1.02620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.203 + 1.39i)T \)
5 \( 1 + (-0.871 - 2.05i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.0775 + 0.0775i)T + 3iT^{2} \)
7 \( 1 + (-2.89 + 2.89i)T - 7iT^{2} \)
11 \( 1 + 3.20iT - 11T^{2} \)
17 \( 1 + (0.217 + 0.217i)T + 17iT^{2} \)
19 \( 1 - 5.01T + 19T^{2} \)
23 \( 1 + (4.02 + 4.02i)T + 23iT^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 2.20iT - 31T^{2} \)
37 \( 1 + (-0.278 - 0.278i)T + 37iT^{2} \)
41 \( 1 - 5.39T + 41T^{2} \)
43 \( 1 + (-1.36 - 1.36i)T + 43iT^{2} \)
47 \( 1 + (4.21 - 4.21i)T - 47iT^{2} \)
53 \( 1 + (0.797 - 0.797i)T - 53iT^{2} \)
59 \( 1 + 9.66T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + (3.68 - 3.68i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (-0.326 + 0.326i)T - 73iT^{2} \)
79 \( 1 - 6.73T + 79T^{2} \)
83 \( 1 + (-2.35 - 2.35i)T + 83iT^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−11.46459084991202419061380160127, −10.92727836417926543968408666628, −10.12519579133529249638847190801, −9.112064048064512541403723595945, −7.913397062722627064857894536724, −6.69949756866038540486525758669, −5.41738690530556409760658907491, −4.05076127352484507079384914096, −3.02004056754264229424138064236, −1.24307998837142495992209153662, 2.03138994582102201932094950638, 4.44149999757032989290627059468, 5.19828047273624754570065570343, 5.87756192334654254259844268625, 7.68476192853100668311472621234, 8.065454829785155204117091984487, 9.231619714065372178182817291207, 9.934464034814973118346994316878, 11.61923861240114992644560212846, 12.31486601323007932508404781848

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