Properties

Label 2-260-20.3-c1-0-15
Degree $2$
Conductor $260$
Sign $-0.403 + 0.915i$
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  + (−0.516 − 1.31i)2-s + (−2.23 + 2.23i)3-s + (−1.46 + 1.36i)4-s + (−2.20 + 0.390i)5-s + (4.10 + 1.79i)6-s + (0.918 + 0.918i)7-s + (2.54 + 1.22i)8-s − 7.03i·9-s + (1.65 + 2.69i)10-s − 3.84i·11-s + (0.236 − 6.33i)12-s + (−0.707 − 0.707i)13-s + (0.734 − 1.68i)14-s + (4.05 − 5.80i)15-s + (0.297 − 3.98i)16-s + (−1.82 + 1.82i)17-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−1.29 + 1.29i)3-s + (−0.732 + 0.680i)4-s + (−0.984 + 0.174i)5-s + (1.67 + 0.731i)6-s + (0.347 + 0.347i)7-s + (0.901 + 0.433i)8-s − 2.34i·9-s + (0.522 + 0.852i)10-s − 1.15i·11-s + (0.0681 − 1.82i)12-s + (−0.196 − 0.196i)13-s + (0.196 − 0.450i)14-s + (1.04 − 1.49i)15-s + (0.0744 − 0.997i)16-s + (−0.442 + 0.442i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.403 + 0.915i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.403 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141874 - 0.217601i\)
\(L(\frac12)\) \(\approx\) \(0.141874 - 0.217601i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.516 + 1.31i)T \)
5 \( 1 + (2.20 - 0.390i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (2.23 - 2.23i)T - 3iT^{2} \)
7 \( 1 + (-0.918 - 0.918i)T + 7iT^{2} \)
11 \( 1 + 3.84iT - 11T^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 + (-4.88 + 4.88i)T - 23iT^{2} \)
29 \( 1 + 8.40iT - 29T^{2} \)
31 \( 1 - 2.11iT - 31T^{2} \)
37 \( 1 + (4.85 - 4.85i)T - 37iT^{2} \)
41 \( 1 - 1.76T + 41T^{2} \)
43 \( 1 + (0.980 - 0.980i)T - 43iT^{2} \)
47 \( 1 + (6.80 + 6.80i)T + 47iT^{2} \)
53 \( 1 + (2.86 + 2.86i)T + 53iT^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 - 2.94T + 61T^{2} \)
67 \( 1 + (-1.81 - 1.81i)T + 67iT^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 + (11.5 + 11.5i)T + 73iT^{2} \)
79 \( 1 - 4.41T + 79T^{2} \)
83 \( 1 + (-4.20 + 4.20i)T - 83iT^{2} \)
89 \( 1 + 9.46iT - 89T^{2} \)
97 \( 1 + (-2.33 + 2.33i)T - 97iT^{2} \)
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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

−11.51654263320859352989254575348, −10.81478379171430633817716022727, −10.25103217974403140852447841065, −8.963345984633721649494109915984, −8.233636402204780054434558937103, −6.48954169307093516589727198026, −5.10289552716156449248655059769, −4.26868216262095684306111137909, −3.22044711442389534584462863899, −0.29593743770894239704323611390, 1.39708846567737464509982060291, 4.54268228435501794997092380006, 5.27935820353568170547605840181, 6.70416565850973830537824481115, 7.24680636492731108221351587691, 7.85267503470771565763062043994, 9.163916396784741292055338532235, 10.68760651524444654190321746792, 11.31532874261053194733693476572, 12.43645561949548075154280214452

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