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Properties

Label 2-1300-65.4-c1-0-3
Degree $2$
Conductor $1300$
Sign $-0.0781 - 0.996i$
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  + (1.38 + 0.800i)3-s + (2.16 + 3.75i)7-s + (−0.219 − 0.380i)9-s + (1.5 + 0.866i)11-s + (1.81 + 3.11i)13-s + (−6.49 + 3.75i)17-s + (−4.65 + 2.68i)19-s + 6.93i·21-s + (1.00 + 0.580i)23-s − 5.50i·27-s + (−1.01 + 1.75i)29-s − 7.86i·31-s + (1.38 + 2.40i)33-s + (4.76 − 8.25i)37-s + (0.0316 + 5.76i)39-s + ⋯
L(s)  = 1  + (0.800 + 0.461i)3-s + (0.818 + 1.41i)7-s + (−0.0732 − 0.126i)9-s + (0.452 + 0.261i)11-s + (0.504 + 0.863i)13-s + (−1.57 + 0.909i)17-s + (−1.06 + 0.616i)19-s + 1.51i·21-s + (0.209 + 0.121i)23-s − 1.05i·27-s + (−0.187 + 0.325i)29-s − 1.41i·31-s + (0.241 + 0.417i)33-s + (0.783 − 1.35i)37-s + (0.00506 + 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0781 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0781 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.0781 - 0.996i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.0781 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.178106046\)
\(L(\frac12)\) \(\approx\) \(2.178106046\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 3.11i)T \)
good3 \( 1 + (-1.38 - 0.800i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.16 - 3.75i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (6.49 - 3.75i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.65 - 2.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.00 - 0.580i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.01 - 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (-4.76 + 8.25i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.69 - 3.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.62 + 2.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 12.6iT - 53T^{2} \)
59 \( 1 + (5.49 - 3.17i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.85 + 3.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.63 + 4.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.8 - 6.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.23T + 73T^{2} \)
79 \( 1 - 8.16T + 79T^{2} \)
83 \( 1 - 0.456T + 83T^{2} \)
89 \( 1 + (-11.4 - 6.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.40 + 2.43i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.384453114404295217966548722949, −9.057650755249495861955878910291, −8.539092399485561051544376868512, −7.70072439089596409397588071539, −6.30191169743054078962747520655, −5.89786604602586559644626270377, −4.38543185567257475904084443732, −4.03870642135082724186806745200, −2.50801347458485083862423814699, −1.90000672681338178520342796832, 0.824475140068356627057962390120, 2.08694892282751372120343049559, 3.15934029235745371670703384935, 4.28119564186903934625755870872, 4.96554986377867160291819706115, 6.41258020137059977727973645518, 7.12109006385074000988574774837, 7.86073946809053948053518955052, 8.544410959073009002071468599792, 9.157279442792331509797199002610

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