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Properties

Label 2-260-20.7-c1-0-20
Degree $2$
Conductor $260$
Sign $0.698 - 0.715i$
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  + (1.32 + 0.498i)2-s + (1.08 + 1.08i)3-s + (1.50 + 1.31i)4-s + (−0.398 − 2.20i)5-s + (0.898 + 1.98i)6-s + (−0.555 + 0.555i)7-s + (1.33 + 2.49i)8-s − 0.626i·9-s + (0.570 − 3.11i)10-s + 2.61i·11-s + (0.198 + 3.07i)12-s + (0.707 − 0.707i)13-s + (−1.01 + 0.457i)14-s + (1.96 − 2.83i)15-s + (0.515 + 3.96i)16-s + (−2.63 − 2.63i)17-s + ⋯
L(s)  = 1  + (0.935 + 0.352i)2-s + (0.628 + 0.628i)3-s + (0.751 + 0.659i)4-s + (−0.178 − 0.984i)5-s + (0.366 + 0.810i)6-s + (−0.209 + 0.209i)7-s + (0.470 + 0.882i)8-s − 0.208i·9-s + (0.180 − 0.983i)10-s + 0.787i·11-s + (0.0574 + 0.887i)12-s + (0.196 − 0.196i)13-s + (−0.270 + 0.122i)14-s + (0.506 − 0.730i)15-s + (0.128 + 0.991i)16-s + (−0.639 − 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.20477 + 0.928570i\)
\(L(\frac12)\) \(\approx\) \(2.20477 + 0.928570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.498i)T \)
5 \( 1 + (0.398 + 2.20i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.08 - 1.08i)T + 3iT^{2} \)
7 \( 1 + (0.555 - 0.555i)T - 7iT^{2} \)
11 \( 1 - 2.61iT - 11T^{2} \)
17 \( 1 + (2.63 + 2.63i)T + 17iT^{2} \)
19 \( 1 + 5.53T + 19T^{2} \)
23 \( 1 + (0.817 + 0.817i)T + 23iT^{2} \)
29 \( 1 + 1.74iT - 29T^{2} \)
31 \( 1 + 1.39iT - 31T^{2} \)
37 \( 1 + (0.838 + 0.838i)T + 37iT^{2} \)
41 \( 1 - 0.268T + 41T^{2} \)
43 \( 1 + (-3.82 - 3.82i)T + 43iT^{2} \)
47 \( 1 + (-8.05 + 8.05i)T - 47iT^{2} \)
53 \( 1 + (-6.61 + 6.61i)T - 53iT^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 9.92T + 61T^{2} \)
67 \( 1 + (4.22 - 4.22i)T - 67iT^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 + (7.67 - 7.67i)T - 73iT^{2} \)
79 \( 1 + 6.13T + 79T^{2} \)
83 \( 1 + (-2.27 - 2.27i)T + 83iT^{2} \)
89 \( 1 - 5.29iT - 89T^{2} \)
97 \( 1 + (3.44 + 3.44i)T + 97iT^{2} \)
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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

−12.40133455363252129053744531518, −11.44753298488909842508604577314, −10.12543014476426234427391453746, −9.024022861875974164100851388717, −8.336692436877610898822472255090, −7.07468243053418445440391930105, −5.85695152505659584801120595963, −4.58615329522924029987100812630, −3.96942781049828369551239794739, −2.43965517709263886111689577747, 2.02991622412712541318387865308, 3.11867645341281419737894860729, 4.23488970616909764341180176326, 5.94899506323147004908082235341, 6.75957915336789702114897729145, 7.70387274284183227074013983415, 8.888130130396203965964208482389, 10.58329139597755232168544253958, 10.78852661283748145555864930707, 12.00496224205893265420290075101

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