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Properties

Label 2-1300-13.4-c1-0-13
Degree $2$
Conductor $1300$
Sign $0.515 + 0.856i$
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  + (0.800 − 1.38i)3-s + (−3.75 + 2.16i)7-s + (0.219 + 0.380i)9-s + (1.5 + 0.866i)11-s + (3.11 − 1.81i)13-s + (−3.75 − 6.49i)17-s + (4.65 − 2.68i)19-s + 6.93i·21-s + (0.580 − 1.00i)23-s + 5.50·27-s + (1.01 − 1.75i)29-s − 7.86i·31-s + (2.40 − 1.38i)33-s + (8.25 + 4.76i)37-s + (−0.0316 − 5.76i)39-s + ⋯
L(s)  = 1  + (0.461 − 0.800i)3-s + (−1.41 + 0.818i)7-s + (0.0732 + 0.126i)9-s + (0.452 + 0.261i)11-s + (0.863 − 0.504i)13-s + (−0.909 − 1.57i)17-s + (1.06 − 0.616i)19-s + 1.51i·21-s + (0.121 − 0.209i)23-s + 1.05·27-s + (0.187 − 0.325i)29-s − 1.41i·31-s + (0.417 − 0.241i)33-s + (1.35 + 0.783i)37-s + (−0.00506 − 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.515 + 0.856i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.743142904\)
\(L(\frac12)\) \(\approx\) \(1.743142904\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 + 1.81i)T \)
good3 \( 1 + (-0.800 + 1.38i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.75 - 2.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.75 + 6.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.65 + 2.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.580 + 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.01 + 1.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (-8.25 - 4.76i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.69 - 3.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.09 + 3.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
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 \( 1 + (-4.55 - 2.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.8 - 6.25i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.23iT - 73T^{2} \)
79 \( 1 + 8.16T + 79T^{2} \)
83 \( 1 + 0.456iT - 83T^{2} \)
89 \( 1 + (11.4 + 6.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.43 + 1.40i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.390908980767957565121624393087, −8.820020186674266151178752712255, −7.81944563192562275292797343014, −7.01446813324479224601581185461, −6.39635660135937349310263219757, −5.50399677611912871439614837921, −4.28608685259527042726744681262, −2.92726470728449935363588952111, −2.49440303927467740528637774654, −0.817167000894114044186961684201, 1.20545508915614206529990880962, 3.00284325544509560854719068558, 3.90128005023982054415122756888, 4.09532040064167323593991681737, 5.75288371967008302822635986182, 6.51731382883726554282630435366, 7.17400808958401509975199373431, 8.482911683777022316709700811376, 9.059129913949889478112987951462, 9.765362317197933880632865805407

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