Properties

Label 2-260-260.207-c1-0-14
Degree $2$
Conductor $260$
Sign $0.904 - 0.427i$
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  + (−1.27 − 0.601i)2-s + (1.55 + 1.55i)3-s + (1.27 + 1.54i)4-s + (1.95 − 1.08i)5-s + (−1.05 − 2.92i)6-s + (3.02 + 3.02i)7-s + (−0.706 − 2.73i)8-s + 1.82i·9-s + (−3.15 + 0.205i)10-s − 4.00·11-s + (−0.409 + 4.37i)12-s + (−0.933 − 3.48i)13-s + (−2.05 − 5.70i)14-s + (4.71 + 1.36i)15-s + (−0.743 + 3.93i)16-s + (−0.753 − 0.753i)17-s + ⋯
L(s)  = 1  + (−0.904 − 0.425i)2-s + (0.896 + 0.896i)3-s + (0.638 + 0.770i)4-s + (0.875 − 0.483i)5-s + (−0.429 − 1.19i)6-s + (1.14 + 1.14i)7-s + (−0.249 − 0.968i)8-s + 0.606i·9-s + (−0.997 + 0.0651i)10-s − 1.20·11-s + (−0.118 + 1.26i)12-s + (−0.258 − 0.965i)13-s + (−0.549 − 1.52i)14-s + (1.21 + 0.351i)15-s + (−0.185 + 0.982i)16-s + (−0.182 − 0.182i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.904 - 0.427i$
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: \((2,\ 260,\ (\ :1/2),\ 0.904 - 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28233 + 0.287593i\)
\(L(\frac12)\) \(\approx\) \(1.28233 + 0.287593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.601i)T \)
5 \( 1 + (-1.95 + 1.08i)T \)
13 \( 1 + (0.933 + 3.48i)T \)
good3 \( 1 + (-1.55 - 1.55i)T + 3iT^{2} \)
7 \( 1 + (-3.02 - 3.02i)T + 7iT^{2} \)
11 \( 1 + 4.00T + 11T^{2} \)
17 \( 1 + (0.753 + 0.753i)T + 17iT^{2} \)
19 \( 1 - 4.01iT - 19T^{2} \)
23 \( 1 + (3.75 + 3.75i)T + 23iT^{2} \)
29 \( 1 - 6.60iT - 29T^{2} \)
31 \( 1 - 2.35T + 31T^{2} \)
37 \( 1 + (0.454 - 0.454i)T - 37iT^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 + (4.32 + 4.32i)T + 43iT^{2} \)
47 \( 1 + (-3.87 - 3.87i)T + 47iT^{2} \)
53 \( 1 + (-1.34 + 1.34i)T - 53iT^{2} \)
59 \( 1 + 0.838iT - 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 + (3.69 + 3.69i)T + 67iT^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (-5.33 - 5.33i)T + 73iT^{2} \)
79 \( 1 + 3.43T + 79T^{2} \)
83 \( 1 + (-4.47 + 4.47i)T - 83iT^{2} \)
89 \( 1 - 6.30T + 89T^{2} \)
97 \( 1 + (-9.64 + 9.64i)T - 97iT^{2} \)
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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

−12.04749267379940251657829326551, −10.54534228937755372560072934102, −10.19781293643616045592718488159, −9.055083650863794571591616572327, −8.510477064169096786927919422144, −7.81390945796863846274443032908, −5.81822630676286412818251083800, −4.75373170673852127244828492348, −2.97400272712245451568766907795, −2.03808965392176013663625205666, 1.59656620099104308174635813733, 2.52388844626297441385722391261, 4.85821632942325936159305479294, 6.33685360272056698894649282631, 7.42114362325684695397322100056, 7.75923971261173218480444515494, 8.836927574841579725904585125691, 9.948003619910104242824346636544, 10.73265494077961190385683872745, 11.62922360246540092783000792701

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