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Properties

Label 2-260-260.3-c1-0-11
Degree $2$
Conductor $260$
Sign $-0.521 - 0.853i$
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  + (−0.0733 + 1.41i)2-s + (2.99 + 0.802i)3-s + (−1.98 − 0.207i)4-s + (−1.68 + 1.47i)5-s + (−1.35 + 4.16i)6-s + (−0.986 + 0.264i)7-s + (0.438 − 2.79i)8-s + (5.71 + 3.30i)9-s + (−1.95 − 2.48i)10-s + (−0.266 + 0.154i)11-s + (−5.78 − 2.21i)12-s + (−1.08 + 3.43i)13-s + (−0.301 − 1.41i)14-s + (−6.21 + 3.06i)15-s + (3.91 + 0.824i)16-s + (−0.295 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.0518 + 0.998i)2-s + (1.72 + 0.463i)3-s + (−0.994 − 0.103i)4-s + (−0.751 + 0.659i)5-s + (−0.552 + 1.70i)6-s + (−0.372 + 0.0999i)7-s + (0.155 − 0.987i)8-s + (1.90 + 1.10i)9-s + (−0.619 − 0.784i)10-s + (−0.0804 + 0.0464i)11-s + (−1.67 − 0.639i)12-s + (−0.301 + 0.953i)13-s + (−0.0804 − 0.377i)14-s + (−1.60 + 0.791i)15-s + (0.978 + 0.206i)16-s + (−0.0717 − 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807869 + 1.43963i\)
\(L(\frac12)\) \(\approx\) \(0.807869 + 1.43963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0733 - 1.41i)T \)
5 \( 1 + (1.68 - 1.47i)T \)
13 \( 1 + (1.08 - 3.43i)T \)
good3 \( 1 + (-2.99 - 0.802i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.986 - 0.264i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.266 - 0.154i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.295 + 1.10i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.40 + 5.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.36 - 1.97i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.81 + 1.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.52iT - 31T^{2} \)
37 \( 1 + (9.73 + 2.60i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.175 - 0.303i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.46 + 5.44i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.04 - 3.04i)T + 47iT^{2} \)
53 \( 1 + (0.101 + 0.101i)T + 53iT^{2} \)
59 \( 1 + (0.416 - 0.721i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.06 + 5.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.145 - 0.542i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-8.04 - 4.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.25 - 5.25i)T + 73iT^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + (0.855 - 0.855i)T - 83iT^{2} \)
89 \( 1 + (10.8 - 6.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.14 - 4.28i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.70822611443151591927993832544, −11.20308314997627522661634008650, −9.862576065826761660106874652172, −9.203160140217172593612626517970, −8.462061288859865444535239060851, −7.31697155488869714255759788345, −6.94238436710337730806677090908, −4.89360463741440035728625388249, −3.83706741276831350463150897468, −2.81577694885339515978018562261, 1.33817597582671699417350481810, 3.02486723609030304462474393837, 3.62532044174864157098906413899, 5.03294084727935408523661827076, 7.22599004757835987894290247175, 8.183989650980818780499073740595, 8.677610349046292675359528941454, 9.645446154825238821689926156742, 10.53913866838775409236192614947, 12.04064622126013076092817624016

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