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Properties

Label 2-260-65.7-c1-0-5
Degree $2$
Conductor $260$
Sign $0.439 + 0.898i$
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  + (0.133 + 0.5i)3-s + (−1 − 2i)5-s + (−1.23 − 2.13i)7-s + (2.36 − 1.36i)9-s + (−1.13 − 4.23i)11-s + (3 + 2i)13-s + (0.866 − 0.767i)15-s + (−0.866 − 0.232i)17-s + (2.86 + 0.767i)19-s + (0.901 − 0.901i)21-s + (−0.133 + 0.0358i)23-s + (−3 + 4i)25-s + (2.09 + 2.09i)27-s + (1.5 + 0.866i)29-s + (−5.19 − 5.19i)31-s + ⋯
L(s)  = 1  + (0.0773 + 0.288i)3-s + (−0.447 − 0.894i)5-s + (−0.465 − 0.806i)7-s + (0.788 − 0.455i)9-s + (−0.341 − 1.27i)11-s + (0.832 + 0.554i)13-s + (0.223 − 0.198i)15-s + (−0.210 − 0.0562i)17-s + (0.657 + 0.176i)19-s + (0.196 − 0.196i)21-s + (−0.0279 + 0.00748i)23-s + (−0.600 + 0.800i)25-s + (0.403 + 0.403i)27-s + (0.278 + 0.160i)29-s + (−0.933 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979688 - 0.611344i\)
\(L(\frac12)\) \(\approx\) \(0.979688 - 0.611344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
13 \( 1 + (-3 - 2i)T \)
good3 \( 1 + (-0.133 - 0.5i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.23 + 2.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.13 + 4.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.866 + 0.232i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.86 - 0.767i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.133 - 0.0358i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.19 + 5.19i)T + 31iT^{2} \)
37 \( 1 + (0.767 - 1.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.33 + 2.5i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.59 - 5.96i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + (-2.46 + 2.46i)T - 53iT^{2} \)
59 \( 1 + (2.33 - 8.69i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.6 - 6.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.598 + 2.23i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 0.535iT - 79T^{2} \)
83 \( 1 - 2.92T + 83T^{2} \)
89 \( 1 + (-14.7 + 3.96i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.69 - 3.86i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.72789434230919282116147036188, −10.93494573566173550215555792615, −9.811984250244785250978925526305, −8.983233782755644365325510729076, −8.012884743024200980223629908603, −6.89046694179901144537616373774, −5.66388492060918873816781367191, −4.25400723953090574799473415763, −3.51646579956256495964805868985, −0.981538439482748637147898898564, 2.15928606466751431862994085829, 3.47450023298770843653119276078, 4.94898726315901955375340244502, 6.33799017358164502769520730446, 7.24702629035058128595045470691, 8.053137253504469657679272294914, 9.401490402465777624673333279854, 10.30966562789672717155515967705, 11.14931895262325886254665440416, 12.34068059202060480467956746157

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