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Properties

Label 2-260-260.103-c1-0-12
Degree $2$
Conductor $260$
Sign $0.688 - 0.725i$
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  + (−1.41 − 0.0324i)2-s + (−2.32 + 2.32i)3-s + (1.99 + 0.0918i)4-s + (2.23 + 0.0191i)5-s + (3.36 − 3.21i)6-s + (1.97 − 1.97i)7-s + (−2.82 − 0.194i)8-s − 7.85i·9-s + (−3.16 − 0.0997i)10-s + 2.76·11-s + (−4.86 + 4.44i)12-s + (3.56 + 0.543i)13-s + (−2.85 + 2.72i)14-s + (−5.25 + 5.16i)15-s + (3.98 + 0.367i)16-s + (−1.79 + 1.79i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0229i)2-s + (−1.34 + 1.34i)3-s + (0.998 + 0.0459i)4-s + (0.999 + 0.00856i)5-s + (1.37 − 1.31i)6-s + (0.745 − 0.745i)7-s + (−0.997 − 0.0688i)8-s − 2.61i·9-s + (−0.999 − 0.0315i)10-s + 0.833·11-s + (−1.40 + 1.28i)12-s + (0.988 + 0.150i)13-s + (−0.761 + 0.727i)14-s + (−1.35 + 1.33i)15-s + (0.995 + 0.0917i)16-s + (−0.436 + 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.688 - 0.725i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.688 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666369 + 0.286207i\)
\(L(\frac12)\) \(\approx\) \(0.666369 + 0.286207i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0324i)T \)
5 \( 1 + (-2.23 - 0.0191i)T \)
13 \( 1 + (-3.56 - 0.543i)T \)
good3 \( 1 + (2.32 - 2.32i)T - 3iT^{2} \)
7 \( 1 + (-1.97 + 1.97i)T - 7iT^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
17 \( 1 + (1.79 - 1.79i)T - 17iT^{2} \)
19 \( 1 + 2.06iT - 19T^{2} \)
23 \( 1 + (-1.07 + 1.07i)T - 23iT^{2} \)
29 \( 1 - 7.19iT - 29T^{2} \)
31 \( 1 + 2.08T + 31T^{2} \)
37 \( 1 + (2.10 + 2.10i)T + 37iT^{2} \)
41 \( 1 + 2.07iT - 41T^{2} \)
43 \( 1 + (-1.37 + 1.37i)T - 43iT^{2} \)
47 \( 1 + (3.03 - 3.03i)T - 47iT^{2} \)
53 \( 1 + (4.80 + 4.80i)T + 53iT^{2} \)
59 \( 1 - 9.65iT - 59T^{2} \)
61 \( 1 - 6.83T + 61T^{2} \)
67 \( 1 + (-10.5 + 10.5i)T - 67iT^{2} \)
71 \( 1 + 3.76T + 71T^{2} \)
73 \( 1 + (-0.228 + 0.228i)T - 73iT^{2} \)
79 \( 1 + 3.36T + 79T^{2} \)
83 \( 1 + (-2.38 - 2.38i)T + 83iT^{2} \)
89 \( 1 - 3.39T + 89T^{2} \)
97 \( 1 + (-5.33 - 5.33i)T + 97iT^{2} \)
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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

−11.47356799048677979005168373759, −10.87209303154602768135098481388, −10.42830009282410040644999659763, −9.366791033998529353220849701076, −8.762045524080141631242466274848, −6.87109458899507643475152616450, −6.16025137841228593496887045059, −5.05781495447777434789214428123, −3.75036698984647477745614417718, −1.26074707487305613188102294956, 1.25032859595786150879155578387, 2.16834705922613248298370353212, 5.30305406478552473234081256476, 6.11369026470566311392256922760, 6.72160487647285814513471842392, 7.929367527337510116954265913175, 8.841111732217017167292275124218, 10.07533599426515328025167025292, 11.24525825594922410091024853253, 11.53349308934140885563284805502

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