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Properties

Label 2-1300-65.37-c1-0-5
Degree $2$
Conductor $1300$
Sign $0.836 + 0.547i$
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  + (−1.86 − 0.5i)3-s + (−2.23 + 3.86i)7-s + (0.633 + 0.366i)9-s + (−2.86 − 0.767i)11-s + (−3 − 2i)13-s + (−0.866 − 3.23i)17-s + (1.13 + 4.23i)19-s + (6.09 − 6.09i)21-s + (1.86 − 6.96i)23-s + (3.09 + 3.09i)27-s + (1.5 − 0.866i)29-s + (5.19 + 5.19i)31-s + (4.96 + 2.86i)33-s + (4.23 + 7.33i)37-s + (4.59 + 5.23i)39-s + ⋯
L(s)  = 1  + (−1.07 − 0.288i)3-s + (−0.843 + 1.46i)7-s + (0.211 + 0.122i)9-s + (−0.864 − 0.231i)11-s + (−0.832 − 0.554i)13-s + (−0.210 − 0.783i)17-s + (0.260 + 0.970i)19-s + (1.33 − 1.33i)21-s + (0.389 − 1.45i)23-s + (0.596 + 0.596i)27-s + (0.278 − 0.160i)29-s + (0.933 + 0.933i)31-s + (0.864 + 0.498i)33-s + (0.695 + 1.20i)37-s + (0.736 + 0.837i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.836 + 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6339460473\)
\(L(\frac12)\) \(\approx\) \(0.6339460473\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.86 + 0.5i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.23 - 3.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.86 + 0.767i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.866 + 3.23i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.13 - 4.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.86 + 6.96i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.5 + 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.19 - 5.19i)T + 31iT^{2} \)
37 \( 1 + (-4.23 - 7.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.669 + 2.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.59 - 0.964i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + (-4.46 + 4.46i)T - 53iT^{2} \)
59 \( 1 + (-6.33 + 1.69i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.6 + 7.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.59 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 1.07iT - 73T^{2} \)
79 \( 1 - 7.46iT - 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (0.794 - 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.69 + 2.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.787207554468611451700374892189, −8.694172048611909580469089607828, −8.045882560508630414133607762638, −6.75006615805784424902556814857, −6.29385727569092581703886292898, −5.31552473077410939917466302273, −4.98529706301078679110551615380, −3.12254211072698910839517168526, −2.45910970323899202999500585790, −0.47940629110845485155655463308, 0.72441295454777303955676667866, 2.57453835038326076197608865736, 3.86784043699476933423043351977, 4.66250572808524824630801150757, 5.49028750611833719412808526437, 6.45849777754220405812307206332, 7.16136909928084442213627405504, 7.83628353767232336210807184851, 9.183546116876726831948778102557, 10.03738973496859890271659386754

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