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Properties

Label 2-1300-65.32-c1-0-0
Degree $2$
Conductor $1300$
Sign $-0.392 + 0.919i$
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  + (0.690 + 2.57i)3-s + (−1.87 + 1.08i)7-s + (−3.56 + 2.06i)9-s + (−5.13 + 1.37i)11-s + (−2.35 − 2.72i)13-s + (−1.07 − 0.288i)17-s + (1.69 − 6.33i)19-s + (−4.08 − 4.08i)21-s + (0.718 − 0.192i)23-s + (−2.11 − 2.11i)27-s + (−0.0866 − 0.0500i)29-s + (3.90 − 3.90i)31-s + (−7.09 − 12.2i)33-s + (−5.91 − 3.41i)37-s + (5.40 − 7.95i)39-s + ⋯
L(s)  = 1  + (0.398 + 1.48i)3-s + (−0.708 + 0.408i)7-s + (−1.18 + 0.686i)9-s + (−1.54 + 0.414i)11-s + (−0.653 − 0.757i)13-s + (−0.260 − 0.0699i)17-s + (0.389 − 1.45i)19-s + (−0.890 − 0.890i)21-s + (0.149 − 0.0401i)23-s + (−0.406 − 0.406i)27-s + (−0.0160 − 0.00929i)29-s + (0.700 − 0.700i)31-s + (−1.23 − 2.13i)33-s + (−0.972 − 0.561i)37-s + (0.866 − 1.27i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.392 + 0.919i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1697254245\)
\(L(\frac12)\) \(\approx\) \(0.1697254245\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.35 + 2.72i)T \)
good3 \( 1 + (-0.690 - 2.57i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.87 - 1.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.13 - 1.37i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.07 + 0.288i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.69 + 6.33i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.718 + 0.192i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.0866 + 0.0500i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.90 + 3.90i)T - 31iT^{2} \)
37 \( 1 + (5.91 + 3.41i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.36 - 5.10i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.959 - 3.57i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 2.04iT - 47T^{2} \)
53 \( 1 + (8.28 - 8.28i)T - 53iT^{2} \)
59 \( 1 + (-12.1 - 3.25i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.66 + 8.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.51 - 9.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.19 + 2.46i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 - 15.3iT - 79T^{2} \)
83 \( 1 - 0.473iT - 83T^{2} \)
89 \( 1 + (0.560 + 2.09i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.34 + 10.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.04679020800898820983888141606, −9.565927014079932147734797798892, −8.819089996692416866622548547833, −7.930101730620797860774255774484, −7.03984946025170457850087569437, −5.72814527144174794944007243787, −5.01143154195737617403089410218, −4.35965737192287604187115562009, −2.90654711949116452430852815845, −2.75601843606776322085545511950, 0.06215021627278914990390745926, 1.60151427089190649719940325941, 2.62637267397143035927477814664, 3.51328500309621953157212044140, 4.97840436771046522391818919015, 5.99650172097072278773232593223, 6.80513164339928731735992461457, 7.44837782751668846124471895564, 8.101176294247075410488874765143, 8.845339543614302734434159806197

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