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Properties

Label 2-368-16.5-c1-0-40
Degree $2$
Conductor $368$
Sign $-0.382 + 0.923i$
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 − i)2-s + (1 − i)3-s − 2i·4-s − 2i·6-s − 4i·7-s + (−2 − 2i)8-s + i·9-s + (4 + 4i)11-s + (−2 − 2i)12-s + (−3 + 3i)13-s + (−4 − 4i)14-s − 4·16-s − 2·17-s + (1 + i)18-s + (−4 − 4i)21-s + 8·22-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.577 − 0.577i)3-s − i·4-s − 0.816i·6-s − 1.51i·7-s + (−0.707 − 0.707i)8-s + 0.333i·9-s + (1.20 + 1.20i)11-s + (−0.577 − 0.577i)12-s + (−0.832 + 0.832i)13-s + (−1.06 − 1.06i)14-s − 16-s − 0.485·17-s + (0.235 + 0.235i)18-s + (−0.872 − 0.872i)21-s + 1.70·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25160 - 1.87316i\)
\(L(\frac12)\) \(\approx\) \(1.25160 - 1.87316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
23 \( 1 - iT \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + (-4 - 4i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
29 \( 1 + (-7 + 7i)T - 29iT^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-4 - 4i)T + 37iT^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + (4 + 4i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (8 + 8i)T + 53iT^{2} \)
59 \( 1 + (-7 - 7i)T + 59iT^{2} \)
61 \( 1 + (-2 + 2i)T - 61iT^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-4 + 4i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−11.36387821674999569325546176916, −10.07198832115740968967108547663, −9.742873861262560381185321382052, −8.272205048388940996844110924949, −7.01431035498464557021768319641, −6.63070554317015348478838123544, −4.57914800205004704045163082110, −4.22671008582789310146663312296, −2.55739915143767689554622403566, −1.39467549267263300822400929644, 2.75923361975828761520560075186, 3.53314278490370372718784437860, 4.88002365486117064789566382443, 5.86070609779474056281393104011, 6.68726371160253003477760300878, 8.141460975365559579415243351109, 8.914501515420554901772898283376, 9.336511776146398879549014842424, 10.95558083309029648444831939764, 12.06405003571749166823012809707

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