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Properties

Label 2-260-260.207-c1-0-3
Degree $2$
Conductor $260$
Sign $-0.898 - 0.438i$
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  + (−0.702 + 1.22i)2-s + (1.63 + 1.63i)3-s + (−1.01 − 1.72i)4-s + (−1.85 + 1.24i)5-s + (−3.15 + 0.856i)6-s + (1.80 + 1.80i)7-s + (2.82 − 0.0296i)8-s + 2.34i·9-s + (−0.229 − 3.15i)10-s − 2.86·11-s + (1.16 − 4.47i)12-s + (−1.91 + 3.05i)13-s + (−3.47 + 0.944i)14-s + (−5.07 − 0.989i)15-s + (−1.95 + 3.49i)16-s + (3.05 + 3.05i)17-s + ⋯
L(s)  = 1  + (−0.496 + 0.867i)2-s + (0.943 + 0.943i)3-s + (−0.506 − 0.862i)4-s + (−0.829 + 0.558i)5-s + (−1.28 + 0.349i)6-s + (0.681 + 0.681i)7-s + (0.999 − 0.0104i)8-s + 0.780i·9-s + (−0.0726 − 0.997i)10-s − 0.864·11-s + (0.336 − 1.29i)12-s + (−0.531 + 0.847i)13-s + (−0.929 + 0.252i)14-s + (−1.30 − 0.255i)15-s + (−0.487 + 0.872i)16-s + (0.739 + 0.739i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.898 - 0.438i$
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: \((2,\ 260,\ (\ :1/2),\ -0.898 - 0.438i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243091 + 1.05221i\)
\(L(\frac12)\) \(\approx\) \(0.243091 + 1.05221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.702 - 1.22i)T \)
5 \( 1 + (1.85 - 1.24i)T \)
13 \( 1 + (1.91 - 3.05i)T \)
good3 \( 1 + (-1.63 - 1.63i)T + 3iT^{2} \)
7 \( 1 + (-1.80 - 1.80i)T + 7iT^{2} \)
11 \( 1 + 2.86T + 11T^{2} \)
17 \( 1 + (-3.05 - 3.05i)T + 17iT^{2} \)
19 \( 1 + 5.05iT - 19T^{2} \)
23 \( 1 + (2.42 + 2.42i)T + 23iT^{2} \)
29 \( 1 - 6.79iT - 29T^{2} \)
31 \( 1 - 7.49T + 31T^{2} \)
37 \( 1 + (-2.43 + 2.43i)T - 37iT^{2} \)
41 \( 1 - 3.73iT - 41T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + 43iT^{2} \)
47 \( 1 + (-7.11 - 7.11i)T + 47iT^{2} \)
53 \( 1 + (2.33 - 2.33i)T - 53iT^{2} \)
59 \( 1 + 4.71iT - 59T^{2} \)
61 \( 1 - 9.15T + 61T^{2} \)
67 \( 1 + (1.42 + 1.42i)T + 67iT^{2} \)
71 \( 1 + 2.80T + 71T^{2} \)
73 \( 1 + (-5.53 - 5.53i)T + 73iT^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 + (-3.28 + 3.28i)T - 83iT^{2} \)
89 \( 1 + 0.690T + 89T^{2} \)
97 \( 1 + (-0.611 + 0.611i)T - 97iT^{2} \)
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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

−12.33023940660766315776238639234, −11.08334428710807524742728842929, −10.27331679622084179957438848442, −9.307346440124177146851343662650, −8.415452995221247643703962277690, −7.85661694341215072344081300819, −6.65199025609995757847826596522, −5.09386478664477996570842959744, −4.20498929568939586091839162677, −2.62643286112913370868709588654, 0.962949218932236580241678889107, 2.50247258303835335516860294644, 3.75004027093847040417281419271, 5.06579648706614356514424814232, 7.43399070121323644112945182568, 7.916336792542197831385027082143, 8.274610239960415420849790243288, 9.709852758250580993943551097453, 10.59378634879977871268270106679, 11.84609918736893572193885775533

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