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Properties

Label 2-368-16.13-c1-0-9
Degree $2$
Conductor $368$
Sign $-0.731 - 0.681i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (1.10 + 0.887i)2-s + (0.631 + 0.631i)3-s + (0.425 + 1.95i)4-s + (−2.74 + 2.74i)5-s + (0.135 + 1.25i)6-s − 1.52i·7-s + (−1.26 + 2.52i)8-s − 2.20i·9-s + (−5.44 + 0.585i)10-s + (−2.02 + 2.02i)11-s + (−0.965 + 1.50i)12-s + (2.71 + 2.71i)13-s + (1.35 − 1.68i)14-s − 3.46·15-s + (−3.63 + 1.66i)16-s + 3.66·17-s + ⋯
L(s)  = 1  + (0.778 + 0.627i)2-s + (0.364 + 0.364i)3-s + (0.212 + 0.977i)4-s + (−1.22 + 1.22i)5-s + (0.0551 + 0.512i)6-s − 0.577i·7-s + (−0.447 + 0.894i)8-s − 0.734i·9-s + (−1.72 + 0.185i)10-s + (−0.609 + 0.609i)11-s + (−0.278 + 0.433i)12-s + (0.753 + 0.753i)13-s + (0.362 − 0.450i)14-s − 0.893·15-s + (−0.909 + 0.415i)16-s + 0.888·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.731 - 0.681i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ -0.731 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653864 + 1.66176i\)
\(L(\frac12)\) \(\approx\) \(0.653864 + 1.66176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 0.887i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-0.631 - 0.631i)T + 3iT^{2} \)
5 \( 1 + (2.74 - 2.74i)T - 5iT^{2} \)
7 \( 1 + 1.52iT - 7T^{2} \)
11 \( 1 + (2.02 - 2.02i)T - 11iT^{2} \)
13 \( 1 + (-2.71 - 2.71i)T + 13iT^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 + (-1.20 - 1.20i)T + 19iT^{2} \)
29 \( 1 + (-2.52 - 2.52i)T + 29iT^{2} \)
31 \( 1 - 2.74T + 31T^{2} \)
37 \( 1 + (4.37 - 4.37i)T - 37iT^{2} \)
41 \( 1 - 7.70iT - 41T^{2} \)
43 \( 1 + (-8.96 + 8.96i)T - 43iT^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 + (7.11 - 7.11i)T - 53iT^{2} \)
59 \( 1 + (-5.81 + 5.81i)T - 59iT^{2} \)
61 \( 1 + (6.37 + 6.37i)T + 61iT^{2} \)
67 \( 1 + (0.0827 + 0.0827i)T + 67iT^{2} \)
71 \( 1 - 6.62iT - 71T^{2} \)
73 \( 1 + 7.91iT - 73T^{2} \)
79 \( 1 - 6.42T + 79T^{2} \)
83 \( 1 + (11.6 + 11.6i)T + 83iT^{2} \)
89 \( 1 + 2.76iT - 89T^{2} \)
97 \( 1 + 6.61T + 97T^{2} \)
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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

−11.88836286112091698990610155580, −10.97794660765196669560705601413, −10.08121250753834210452631301938, −8.675452362437172014119323119940, −7.70569179779720876992325235466, −7.06160292293917870811064049054, −6.19435788348049679996988094711, −4.52854637059421615871476423970, −3.71695112095465242551714340924, −2.98487059449252050967082662796, 0.983335368265266889618073751129, 2.74559376709046905725743728081, 3.87637154399835357483845979744, 5.07898179216185446478130753850, 5.73419072072003425153551368408, 7.52322516862238707816996653425, 8.259384874211575978546265451813, 9.060522937713866997663256375865, 10.46536035536175299035096825456, 11.28959214732605859443085029322

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