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Properties

Label 2-1300-65.58-c1-0-14
Degree $2$
Conductor $1300$
Sign $0.547 + 0.836i$
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  + (0.5 − 0.133i)3-s + (−0.133 − 0.232i)7-s + (−2.36 + 1.36i)9-s + (4.59 − 1.23i)11-s + (−2 − 3i)13-s + (0.767 − 2.86i)17-s + (0.866 − 3.23i)19-s + (−0.0980 − 0.0980i)21-s + (0.0358 + 0.133i)23-s + (−2.09 + 2.09i)27-s + (1.03 + 0.598i)29-s + (2.26 − 2.26i)31-s + (2.13 − 1.23i)33-s + (1.86 − 3.23i)37-s + (−1.40 − 1.23i)39-s + ⋯
L(s)  = 1  + (0.288 − 0.0773i)3-s + (−0.0506 − 0.0877i)7-s + (−0.788 + 0.455i)9-s + (1.38 − 0.371i)11-s + (−0.554 − 0.832i)13-s + (0.186 − 0.695i)17-s + (0.198 − 0.741i)19-s + (−0.0214 − 0.0214i)21-s + (0.00748 + 0.0279i)23-s + (−0.403 + 0.403i)27-s + (0.192 + 0.111i)29-s + (0.407 − 0.407i)31-s + (0.371 − 0.214i)33-s + (0.306 − 0.531i)37-s + (−0.224 − 0.197i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.547 + 0.836i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.689783620\)
\(L(\frac12)\) \(\approx\) \(1.689783620\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 3i)T \)
good3 \( 1 + (-0.5 + 0.133i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.133 + 0.232i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.59 + 1.23i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.767 + 2.86i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.866 + 3.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0358 - 0.133i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.03 - 0.598i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.26 + 2.26i)T - 31iT^{2} \)
37 \( 1 + (-1.86 + 3.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.86 + 6.96i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.96 - 2.66i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.46T + 47T^{2} \)
53 \( 1 + (-8.46 - 8.46i)T + 53iT^{2} \)
59 \( 1 + (13.7 + 3.69i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.86 - 0.767i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 0.928iT - 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (-0.794 - 2.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.13 - 1.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.166633513536393044734372748200, −8.997683413325022241228626319057, −7.81132294469603213865517673639, −7.23912781104769724512233302606, −6.14244634781796097174162653011, −5.38958447625028017282321185429, −4.33522995104791983786303262190, −3.23204850639529426093298874515, −2.38060152960421022031031006421, −0.74334641785042497390203144275, 1.35921956322910521343344334389, 2.62723058252487468202790168619, 3.76540583735124761867465640936, 4.45297523352078637322337427233, 5.78563898306963835035772386326, 6.42443340858370305290149878451, 7.30740319050932493114862058848, 8.307334501040540851354438882598, 9.059063993960680621219671319818, 9.574842033975047059864423181489

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