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Properties

Label 2-368-368.229-c0-0-0
Degree $2$
Conductor $368$
Sign $-0.608 - 0.793i$
Analytic cond. $0.183655$
Root an. cond. $0.428550$
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  + (0.866 + 0.5i)2-s + (−1.36 + 1.36i)3-s + (0.499 + 0.866i)4-s + (−1.86 + 0.499i)6-s + 0.999i·8-s − 2.73i·9-s + (−1.86 − 0.5i)12-s + (−0.366 + 0.366i)13-s + (−0.5 + 0.866i)16-s + (1.36 − 2.36i)18-s − i·23-s + (−1.36 − 1.36i)24-s + i·25-s + (−0.5 + 0.133i)26-s + (2.36 + 2.36i)27-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−1.36 + 1.36i)3-s + (0.499 + 0.866i)4-s + (−1.86 + 0.499i)6-s + 0.999i·8-s − 2.73i·9-s + (−1.86 − 0.5i)12-s + (−0.366 + 0.366i)13-s + (−0.5 + 0.866i)16-s + (1.36 − 2.36i)18-s − i·23-s + (−1.36 − 1.36i)24-s + i·25-s + (−0.5 + 0.133i)26-s + (2.36 + 2.36i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(0.183655\)
Root analytic conductor: \(0.428550\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :0),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8585543076\)
\(L(\frac12)\) \(\approx\) \(0.8585543076\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + iT \)
good3 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1 + i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.84577403329638072922851702486, −11.25770972182890629918581402445, −10.29947173645203328554770879489, −9.444364974435847447017775702610, −8.157108208887940525390390181439, −6.59746693296087078687770071644, −6.13186702107194748943957782726, −4.83896419685708902627198099229, −4.50810813480905556858733441861, −3.17031552643764930549474406150, 1.27077102013641717278252256026, 2.70041638883036117255442800914, 4.65305820465626361171054464071, 5.46743808058966314600860001143, 6.39925754598870308255760029752, 7.06192727734637089377886540516, 8.185301885508100416658110641623, 10.03994985082185261853830307921, 10.73854943674617984203364262255, 11.66555195753244977344336961621

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