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Properties

Label 2-1300-65.63-c1-0-13
Degree $2$
Conductor $1300$
Sign $0.503 + 0.863i$
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 − 1.86i)3-s + (3.86 + 2.23i)7-s + (−0.633 − 0.366i)9-s + (−2.86 − 0.767i)11-s + (2 − 3i)13-s + (−3.23 + 0.866i)17-s + (−1.13 − 4.23i)19-s + (6.09 − 6.09i)21-s + (6.96 + 1.86i)23-s + (3.09 − 3.09i)27-s + (−1.5 + 0.866i)29-s + (5.19 + 5.19i)31-s + (−2.86 + 4.96i)33-s + (7.33 − 4.23i)37-s + (−4.59 − 5.23i)39-s + ⋯
L(s)  = 1  + (0.288 − 1.07i)3-s + (1.46 + 0.843i)7-s + (−0.211 − 0.122i)9-s + (−0.864 − 0.231i)11-s + (0.554 − 0.832i)13-s + (−0.783 + 0.210i)17-s + (−0.260 − 0.970i)19-s + (1.33 − 1.33i)21-s + (1.45 + 0.389i)23-s + (0.596 − 0.596i)27-s + (−0.278 + 0.160i)29-s + (0.933 + 0.933i)31-s + (−0.498 + 0.864i)33-s + (1.20 − 0.695i)37-s + (−0.736 − 0.837i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.503 + 0.863i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.178410787\)
\(L(\frac12)\) \(\approx\) \(2.178410787\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 1.86i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.86 - 2.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.86 + 0.767i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.23 - 0.866i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.13 + 4.23i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-6.96 - 1.86i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.5 - 0.866i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.19 - 5.19i)T + 31iT^{2} \)
37 \( 1 + (-7.33 + 4.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.669 + 2.5i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.964 + 3.59i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 2.92iT - 47T^{2} \)
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 \( 1 + (7.86 + 13.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.59 - 1.23i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 + 7.46iT - 79T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (-0.794 + 2.96i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.13 - 3.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.133328263743802214654954053174, −8.538118729436342028486841544231, −7.918519616159546133481419775117, −7.27409062636649656145730877360, −6.25693339878267570067354880501, −5.28589589346096637769712536790, −4.62650571098014960213627118732, −2.92296121702365548283170120511, −2.17361187695418897221548400929, −1.05163686308868157400305503841, 1.34540727165036218867727149516, 2.69833800552389329398593461356, 4.13188842430656132952744746955, 4.41780345790753144114569873167, 5.24857618019769452189515488495, 6.54453131215009765367528465809, 7.51037020957789738906024411937, 8.270978152472517956485655065325, 8.960353726207358391784056848097, 9.919418158893322501381074058652

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