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Properties

Label 2-368-368.229-c0-0-1
Degree $2$
Conductor $368$
Sign $0.991 - 0.130i$
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 + (0.366 − 0.366i)3-s + (0.499 − 0.866i)4-s + (−0.133 + 0.5i)6-s + 0.999i·8-s + 0.732i·9-s + (−0.133 − 0.499i)12-s + (1.36 − 1.36i)13-s + (−0.5 − 0.866i)16-s + (−0.366 − 0.633i)18-s − i·23-s + (0.366 + 0.366i)24-s + i·25-s + (−0.499 + 1.86i)26-s + (0.633 + 0.633i)27-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.366 − 0.366i)3-s + (0.499 − 0.866i)4-s + (−0.133 + 0.5i)6-s + 0.999i·8-s + 0.732i·9-s + (−0.133 − 0.499i)12-s + (1.36 − 1.36i)13-s + (−0.5 − 0.866i)16-s + (−0.366 − 0.633i)18-s − i·23-s + (0.366 + 0.366i)24-s + i·25-s + (−0.499 + 1.86i)26-s + (0.633 + 0.633i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.991 - 0.130i$
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: \((2,\ 368,\ (\ :0),\ 0.991 - 0.130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6473200362\)
\(L(\frac12)\) \(\approx\) \(0.6473200362\)
\(L(1)\) 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 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
29 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
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

−11.10025158620424430645550151522, −10.82801884634297712072593730078, −9.681726987500177409619191931476, −8.602863527232760458046152685760, −8.051453227933110312098111571159, −7.17476858582262702139087525377, −6.04188186891422268217424830116, −5.10217853641352127499414866638, −3.17016564172216996220731066541, −1.61530277404497518509584703417, 1.73662159237131722606297239654, 3.38746497867125279534496855041, 4.14735994161470851306565617641, 6.09937503521079588183333352657, 7.02235708913330005995240628311, 8.217042804921586586946965417214, 9.096226774102062502889232503216, 9.515794735118409278453335343667, 10.68445499292826896057307790900, 11.44419481145073010676614792395

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