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Properties

Label 2-1300-65.28-c1-0-6
Degree $2$
Conductor $1300$
Sign $0.439 - 0.898i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
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 + (1.23 − 2.13i)7-s + (2.36 + 1.36i)9-s + (−1.13 + 4.23i)11-s + (−3 + 2i)13-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 + (−2.09 + 2.09i)27-s + (1.5 − 0.866i)29-s + (−5.19 + 5.19i)31-s + (−1.96 − 1.13i)33-s + (0.767 + 1.33i)37-s + (−0.598 − 1.76i)39-s + ⋯
L(s)  = 1  + (−0.0773 + 0.288i)3-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 + (0.210 − 0.0562i)17-s + (0.657 − 0.176i)19-s + (0.196 + 0.196i)21-s + (0.0279 + 0.00748i)23-s + (−0.403 + 0.403i)27-s + (0.278 − 0.160i)29-s + (−0.933 + 0.933i)31-s + (−0.341 − 0.197i)33-s + (0.126 + 0.218i)37-s + (−0.0957 − 0.283i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.629201733\)
\(L(\frac12)\) \(\approx\) \(1.629201733\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−9.779232263357344744351310894105, −9.294842916082888567680219889633, −7.86504250648383387388730558702, −7.41115279866403116890666462342, −6.78644918521404192620605937235, −5.30900235610228006855284002076, −4.64211245670428001385280665136, −4.01924929078844714293162948356, −2.50734140609112028949136513491, −1.37344298068858266299864707135, 0.75126754263626610031183877254, 2.19781553771638504600090554651, 3.23912441606525379005449547899, 4.37727292081576927971602155173, 5.60652547307677826228663423339, 5.88189062596241072905320823566, 7.29162549529381987607501897963, 7.72453756279040428773516358404, 8.772826861504914840660882098743, 9.371363085060840895662843005246

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