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Properties

Label 2-1300-65.32-c1-0-9
Degree $2$
Conductor $1300$
Sign $-0.00885 - 0.999i$
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  + (0.563 + 2.10i)3-s + (−1.25 + 0.724i)7-s + (−1.50 + 0.867i)9-s + (4.39 − 1.17i)11-s + (3.25 + 1.55i)13-s + (5.20 + 1.39i)17-s + (0.379 − 1.41i)19-s + (−2.23 − 2.23i)21-s + (−3.61 + 0.968i)23-s + (1.94 + 1.94i)27-s + (0.251 + 0.145i)29-s + (−1.02 + 1.02i)31-s + (4.94 + 8.56i)33-s + (−6.21 − 3.59i)37-s + (−1.43 + 7.71i)39-s + ⋯
L(s)  = 1  + (0.325 + 1.21i)3-s + (−0.474 + 0.273i)7-s + (−0.500 + 0.289i)9-s + (1.32 − 0.354i)11-s + (0.902 + 0.431i)13-s + (1.26 + 0.338i)17-s + (0.0870 − 0.324i)19-s + (−0.486 − 0.486i)21-s + (−0.753 + 0.202i)23-s + (0.374 + 0.374i)27-s + (0.0467 + 0.0269i)29-s + (−0.184 + 0.184i)31-s + (0.861 + 1.49i)33-s + (−1.02 − 0.590i)37-s + (−0.230 + 1.23i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00885 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00885 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.00885 - 0.999i$
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: \((2,\ 1300,\ (\ :1/2),\ -0.00885 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.010967761\)
\(L(\frac12)\) \(\approx\) \(2.010967761\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.25 - 1.55i)T \)
good3 \( 1 + (-0.563 - 2.10i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.25 - 0.724i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.39 + 1.17i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-5.20 - 1.39i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.379 + 1.41i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.61 - 0.968i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.251 - 0.145i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.02 - 1.02i)T - 31iT^{2} \)
37 \( 1 + (6.21 + 3.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.53 - 9.46i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.36 + 8.82i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 4.46iT - 47T^{2} \)
53 \( 1 + (3.60 - 3.60i)T - 53iT^{2} \)
59 \( 1 + (5.41 + 1.45i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.00 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.739 - 1.28i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.0 + 2.96i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 5.42iT - 79T^{2} \)
83 \( 1 + 12.0iT - 83T^{2} \)
89 \( 1 + (3.62 + 13.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.38 + 9.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.770085951342303319239459581993, −9.076295029016432217139725121129, −8.649867364530530992976220619364, −7.46587841665149599075924886695, −6.34672775207480299853845346046, −5.73569193921291905865312226083, −4.50396733046461883898686262716, −3.73640398211598217288151986436, −3.15962541507095515857487252922, −1.42254070528595897190786670910, 0.946548241421064016477877057403, 1.86163527865705909867455311262, 3.24081955595068361775010316053, 4.01718836291120153152800052731, 5.47506175599879880603384641637, 6.42606093633203305548206626016, 6.91124854880650331694788169649, 7.83401482595366749052528706971, 8.403234875894712084053598707816, 9.446525574414714554607582358171

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