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Properties

Label 2-1300-65.29-c1-0-15
Degree $2$
Conductor $1300$
Sign $0.435 + 0.900i$
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  + (0.866 − 0.5i)3-s + (−0.866 − 0.5i)7-s + (−1 + 1.73i)9-s + (−1.5 − 2.59i)11-s + (3.46 + i)13-s + (−2.59 − 1.5i)17-s + (2.5 − 4.33i)19-s − 0.999·21-s + (7.79 − 4.5i)23-s + 5i·27-s + (−4.5 − 7.79i)29-s + 8·31-s + (−2.59 − 1.5i)33-s + (6.06 − 3.5i)37-s + (3.49 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.327 − 0.188i)7-s + (−0.333 + 0.577i)9-s + (−0.452 − 0.783i)11-s + (0.960 + 0.277i)13-s + (−0.630 − 0.363i)17-s + (0.573 − 0.993i)19-s − 0.218·21-s + (1.62 − 0.938i)23-s + 0.962i·27-s + (−0.835 − 1.44i)29-s + 1.43·31-s + (−0.452 − 0.261i)33-s + (0.996 − 0.575i)37-s + (0.560 − 0.138i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.759896225\)
\(L(\frac12)\) \(\approx\) \(1.759896225\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.46 - i)T \)
good3 \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.59 + 1.5i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.79 + 4.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.5 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.7 - 8.5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.267912273423956962118449925594, −8.694608493844268274741295418313, −7.986565316582080052323116084771, −7.09095689449418984383890398825, −6.28898911549912719082830747829, −5.30712350416820784321340933428, −4.32459329159153190161803735358, −3.08570589663867033184240462309, −2.42730079612562185956952118107, −0.75145703867598346406042132640, 1.36923270011448134348919816524, 2.88779725957786019793351120341, 3.51120347424117226576223151823, 4.61767005876987824465491350123, 5.66204158984023124221803999088, 6.46555796409089236170214565390, 7.42320356268734416511815933979, 8.319635760163802970780697917132, 9.030933520640172501506698078299, 9.667491150852004848441853758242

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