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Properties

Label 2-368-368.229-c0-0-2
Degree $2$
Conductor $368$
Sign $-0.382 + 0.923i$
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  − i·2-s + (1 − i)3-s − 4-s + (−1 − i)6-s + i·8-s − i·9-s + (−1 + i)12-s + (−1 + i)13-s + 16-s − 18-s − i·23-s + (1 + i)24-s + i·25-s + (1 + i)26-s + (−1 + i)29-s + ⋯
L(s)  = 1  − i·2-s + (1 − i)3-s − 4-s + (−1 − i)6-s + i·8-s − i·9-s + (−1 + i)12-s + (−1 + i)13-s + 16-s − 18-s − i·23-s + (1 + i)24-s + i·25-s + (1 + i)26-s + (−1 + i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.382 + 0.923i$
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: \((2,\ 368,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9602784148\)
\(L(\frac12)\) \(\approx\) \(0.9602784148\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.47801578832473025370070573525, −10.48642594060146291562550307604, −9.285404190152028619638844553390, −8.832928555136649763265159966721, −7.69790227231917311936749284203, −6.91320423440403641386906106046, −5.24984044923540213582331638384, −3.92173147148021763238424341617, −2.66439918347001726091211790418, −1.74348334154485590414732020256, 2.89760098781900648712615012659, 4.05160507481537054614070851378, 4.98097598267395051054219622754, 6.09404278525061925923110168166, 7.56044246992804208335755359480, 8.088331415801705837513086042720, 9.198323329426612619326075372760, 9.741326581684128073761556796810, 10.52811521856243914520236436704, 12.06266416961279480184273815918

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