Properties

Label 2-260-20.3-c1-0-12
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: $\chi_{260} (183, \cdot )$
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

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

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