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Properties

Label 2-368-368.45-c2-0-28
Degree $2$
Conductor $368$
Sign $0.890 - 0.455i$
Analytic cond. $10.0272$
Root an. cond. $3.16658$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.728 − 1.86i)2-s + (4.24 + 4.24i)3-s + (−2.93 − 2.71i)4-s + (10.9 − 4.81i)6-s + (−7.19 + 3.50i)8-s + 26.9i·9-s + (−0.961 − 23.9i)12-s + (18.0 + 18.0i)13-s + (1.28 + 15.9i)16-s + (50.2 + 19.6i)18-s − 23i·23-s + (−45.3 − 15.6i)24-s − 25i·25-s + (46.8 − 20.5i)26-s + (−76.3 + 76.3i)27-s + ⋯
L(s)  = 1  + (0.364 − 0.931i)2-s + (1.41 + 1.41i)3-s + (−0.734 − 0.678i)4-s + (1.83 − 0.802i)6-s + (−0.899 + 0.437i)8-s + 2.99i·9-s + (−0.0801 − 1.99i)12-s + (1.39 + 1.39i)13-s + (0.0800 + 0.996i)16-s + (2.79 + 1.09i)18-s − i·23-s + (−1.89 − 0.652i)24-s − i·25-s + (1.80 − 0.789i)26-s + (−2.82 + 2.82i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(10.0272\)
Root analytic conductor: \(3.16658\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.87235 + 0.692047i\)
\(L(\frac12)\) \(\approx\) \(2.87235 + 0.692047i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 + 1.86i)T \)
23 \( 1 + 23iT \)
good3 \( 1 + (-4.24 - 4.24i)T + 9iT^{2} \)
5 \( 1 + 25iT^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 + 121iT^{2} \)
13 \( 1 + (-18.0 - 18.0i)T + 169iT^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 361iT^{2} \)
29 \( 1 + (19.2 + 19.2i)T + 841iT^{2} \)
31 \( 1 - 51.8T + 961T^{2} \)
37 \( 1 + 1.36e3iT^{2} \)
41 \( 1 + 8.78iT - 1.68e3T^{2} \)
43 \( 1 + 1.84e3iT^{2} \)
47 \( 1 - 42.8T + 2.20e3T^{2} \)
53 \( 1 + 2.80e3iT^{2} \)
59 \( 1 + (70.5 - 70.5i)T - 3.48e3iT^{2} \)
61 \( 1 - 3.72e3iT^{2} \)
67 \( 1 - 4.48e3iT^{2} \)
71 \( 1 + 113. iT - 5.04e3T^{2} \)
73 \( 1 + 144. iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 6.88e3iT^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−10.95349157596742311766981713887, −10.37813657218412437786557562579, −9.443008170784517564028717676996, −8.837541948141802932290352538408, −8.109188449071803104911029505575, −6.17259848245072627394369037423, −4.60365246732489395511600121163, −4.15333803793467841470960646214, −3.08372275969835716822893033573, −1.98729778230869552162176825329, 1.14503111392685338080271710517, 2.99252909878181053484818739920, 3.70751309216358175933559570160, 5.64299050698238389452855214268, 6.52779102945890587885836757935, 7.47830645660638363156382242655, 8.127234313583580731638874623413, 8.765396838305165766769173875230, 9.704731043138905667695051614197, 11.46699629277252798093188333870

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