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Properties

Label 2-2368-37.29-c0-0-0
Degree $2$
Conductor $2368$
Sign $0.849 - 0.527i$
Analytic cond. $1.18178$
Root an. cond. $1.08709$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.133 + 0.5i)5-s + (−0.5 − 0.866i)9-s + (0.366 + 1.36i)13-s + (0.5 + 0.133i)17-s + (0.633 − 0.366i)25-s + (1.36 + 1.36i)29-s + (−0.866 − 0.5i)37-s + (0.866 + 0.5i)41-s + (0.366 − 0.366i)45-s + (0.5 + 0.866i)49-s + (0.5 − 0.133i)61-s + (−0.633 + 0.366i)65-s − 2i·73-s + (−0.499 + 0.866i)81-s + 0.267i·85-s + ⋯
L(s)  = 1  + (0.133 + 0.5i)5-s + (−0.5 − 0.866i)9-s + (0.366 + 1.36i)13-s + (0.5 + 0.133i)17-s + (0.633 − 0.366i)25-s + (1.36 + 1.36i)29-s + (−0.866 − 0.5i)37-s + (0.866 + 0.5i)41-s + (0.366 − 0.366i)45-s + (0.5 + 0.866i)49-s + (0.5 − 0.133i)61-s + (−0.633 + 0.366i)65-s − 2i·73-s + (−0.499 + 0.866i)81-s + 0.267i·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2368\)    =    \(2^{6} \cdot 37\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(1.18178\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2368} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2368,\ (\ :0),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217659796\)
\(L(\frac12)\) \(\approx\) \(1.217659796\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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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

−9.052871610027263805396891751996, −8.721275137411225300686575174427, −7.63626143539688289601991536279, −6.65500288518680440666489689999, −6.41364653703063929561787968848, −5.35495687932921285806447063242, −4.35074100007317798442891906872, −3.45715746841100539725057041193, −2.62568362019767200488616742577, −1.30213411808861013678939666555, 0.976047633387956980358666222851, 2.39708669047457214848956738190, 3.23052893581528314855800798049, 4.39036171899018374201756507513, 5.31981443443012774411681677166, 5.70423921123268091581524789784, 6.79787963631817124110690925590, 7.79014850185634084546287342621, 8.293949334103705111581292277438, 8.915166347175529766164305019421

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