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Properties

Label 2-260-65.32-c1-0-5
Degree $2$
Conductor $260$
Sign $-0.758 + 0.651i$
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  + (−0.690 − 2.57i)3-s + (−0.0143 − 2.23i)5-s + (1.87 − 1.08i)7-s + (−3.56 + 2.06i)9-s + (−5.13 + 1.37i)11-s + (2.35 + 2.72i)13-s + (−5.75 + 1.58i)15-s + (1.07 + 0.288i)17-s + (1.69 − 6.33i)19-s + (−4.08 − 4.08i)21-s + (−0.718 + 0.192i)23-s + (−4.99 + 0.0642i)25-s + (2.11 + 2.11i)27-s + (−0.0866 − 0.0500i)29-s + (3.90 − 3.90i)31-s + ⋯
L(s)  = 1  + (−0.398 − 1.48i)3-s + (−0.00642 − 0.999i)5-s + (0.708 − 0.408i)7-s + (−1.18 + 0.686i)9-s + (−1.54 + 0.414i)11-s + (0.653 + 0.757i)13-s + (−1.48 + 0.408i)15-s + (0.260 + 0.0699i)17-s + (0.389 − 1.45i)19-s + (−0.890 − 0.890i)21-s + (−0.149 + 0.0401i)23-s + (−0.999 + 0.0128i)25-s + (0.406 + 0.406i)27-s + (−0.0160 − 0.00929i)29-s + (0.700 − 0.700i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.758 + 0.651i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361581 - 0.975699i\)
\(L(\frac12)\) \(\approx\) \(0.361581 - 0.975699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0143 + 2.23i)T \)
13 \( 1 + (-2.35 - 2.72i)T \)
good3 \( 1 + (0.690 + 2.57i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.87 + 1.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.13 - 1.37i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.07 - 0.288i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.69 + 6.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.718 - 0.192i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.0866 + 0.0500i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 3.90i)T - 31iT^{2} \)
37 \( 1 + (-5.91 - 3.41i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.36 - 5.10i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.959 + 3.57i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 2.04iT - 47T^{2} \)
53 \( 1 + (-8.28 + 8.28i)T - 53iT^{2} \)
59 \( 1 + (-12.1 - 3.25i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.66 + 8.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.51 + 9.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.19 + 2.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 + 0.473iT - 83T^{2} \)
89 \( 1 + (0.560 + 2.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.34 - 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.70726128101976554621166470965, −11.06892477886650438794163898499, −9.637355499246376594206647778389, −8.286019843429546265851137637572, −7.77382907790210822282567922535, −6.74750048591417847188562503651, −5.50733370026804289990356629978, −4.55963473063037451931166206878, −2.25022088957378147249456800980, −0.894209082720682684795077144650, 2.81062327848751601645914527538, 3.91637323567756690108768324241, 5.35018755543657942468896677645, 5.84142180402501049529434164749, 7.65939568917624600085497242589, 8.495025060875984224491051505672, 9.948139891170963919729869072117, 10.47163089362553799406860748327, 11.05335105072525117621242385833, 12.00063766447907505131664887321

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