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Properties

Label 2-260-65.49-c1-0-4
Degree $2$
Conductor $260$
Sign $0.631 + 0.775i$
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  + (−1.69 + 0.979i)3-s + (−2.16 − 0.571i)5-s + (1.42 − 2.47i)7-s + (0.419 − 0.726i)9-s + (4.84 − 2.79i)11-s + (2.08 − 2.94i)13-s + (4.22 − 1.14i)15-s + (3.40 + 1.96i)17-s + (−7.09 − 4.09i)19-s + 5.59i·21-s + (0.309 − 0.178i)23-s + (4.34 + 2.46i)25-s − 4.23i·27-s + (1.90 + 3.30i)29-s − 8.47i·31-s + ⋯
L(s)  = 1  + (−0.979 + 0.565i)3-s + (−0.966 − 0.255i)5-s + (0.539 − 0.935i)7-s + (0.139 − 0.242i)9-s + (1.46 − 0.843i)11-s + (0.577 − 0.816i)13-s + (1.09 − 0.296i)15-s + (0.824 + 0.476i)17-s + (−1.62 − 0.939i)19-s + 1.22i·21-s + (0.0644 − 0.0372i)23-s + (0.869 + 0.493i)25-s − 0.814i·27-s + (0.354 + 0.613i)29-s − 1.52i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.631 + 0.775i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.631 + 0.775i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712580 - 0.338480i\)
\(L(\frac12)\) \(\approx\) \(0.712580 - 0.338480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.16 + 0.571i)T \)
13 \( 1 + (-2.08 + 2.94i)T \)
good3 \( 1 + (1.69 - 0.979i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.42 + 2.47i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.84 + 2.79i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.40 - 1.96i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.09 + 4.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.309 + 0.178i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.90 - 3.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.47iT - 31T^{2} \)
37 \( 1 + (3.77 + 6.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.85 - 3.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.89 + 1.67i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.64T + 47T^{2} \)
53 \( 1 + 0.581iT - 53T^{2} \)
59 \( 1 + (0.510 + 0.294i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.90 - 5.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.09 - 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.11 - 1.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.16T + 73T^{2} \)
79 \( 1 - 9.85T + 79T^{2} \)
83 \( 1 + 6.09T + 83T^{2} \)
89 \( 1 + (-0.510 + 0.294i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.35 + 4.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54527812216752941582966691758, −11.00236074212129416421391778111, −10.39286613318055886461874739478, −8.833017556058452542538483982387, −8.042251253601035908383758420095, −6.77011651353001994014385613089, −5.67020804035884304504564350392, −4.40562377099175773590570357884, −3.73182323528670655386798595490, −0.77780491296089779557719760760, 1.62540908557097953348772088492, 3.75516515938871056789192356960, 4.96582846757199016803180528706, 6.35654046592619032396790178560, 6.86390635145168779744472526552, 8.229899095836791614105486032559, 9.071563594300103706326987370568, 10.53263502690151368516319082714, 11.55474464855333385272086620022, 12.14923168254063736841631524564

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