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Properties

Label 2-368-92.91-c3-0-15
Degree $2$
Conductor $368$
Sign $0.0227 - 0.999i$
Analytic cond. $21.7127$
Root an. cond. $4.65968$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11i·3-s + 21.1i·5-s + 26.3·7-s + 25.7·9-s − 33.5·11-s + 65.8·13-s − 23.5·15-s + 24.9i·17-s + 100.·19-s + 29.3i·21-s + (96.7 − 52.9i)23-s − 321.·25-s + 58.7i·27-s − 110.·29-s − 238. i·31-s + ⋯
L(s)  = 1  + 0.214i·3-s + 1.88i·5-s + 1.42·7-s + 0.954·9-s − 0.918·11-s + 1.40·13-s − 0.404·15-s + 0.356i·17-s + 1.21·19-s + 0.305i·21-s + (0.877 − 0.480i)23-s − 2.57·25-s + 0.418i·27-s − 0.705·29-s − 1.37i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0227 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0227 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.0227 - 0.999i$
Analytic conductor: \(21.7127\)
Root analytic conductor: \(4.65968\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :3/2),\ 0.0227 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.446379948\)
\(L(\frac12)\) \(\approx\) \(2.446379948\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 + (-96.7 + 52.9i)T \)
good3 \( 1 - 1.11iT - 27T^{2} \)
5 \( 1 - 21.1iT - 125T^{2} \)
7 \( 1 - 26.3T + 343T^{2} \)
11 \( 1 + 33.5T + 1.33e3T^{2} \)
13 \( 1 - 65.8T + 2.19e3T^{2} \)
17 \( 1 - 24.9iT - 4.91e3T^{2} \)
19 \( 1 - 100.T + 6.85e3T^{2} \)
29 \( 1 + 110.T + 2.43e4T^{2} \)
31 \( 1 + 238. iT - 2.97e4T^{2} \)
37 \( 1 - 200. iT - 5.06e4T^{2} \)
41 \( 1 + 309.T + 6.89e4T^{2} \)
43 \( 1 - 17.5T + 7.95e4T^{2} \)
47 \( 1 - 387. iT - 1.03e5T^{2} \)
53 \( 1 - 65.6iT - 1.48e5T^{2} \)
59 \( 1 - 267. iT - 2.05e5T^{2} \)
61 \( 1 + 202. iT - 2.26e5T^{2} \)
67 \( 1 - 480.T + 3.00e5T^{2} \)
71 \( 1 + 598. iT - 3.57e5T^{2} \)
73 \( 1 - 106.T + 3.89e5T^{2} \)
79 \( 1 - 18.4T + 4.93e5T^{2} \)
83 \( 1 + 414.T + 5.71e5T^{2} \)
89 \( 1 + 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 603. iT - 9.12e5T^{2} \)
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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

−10.96625535806891777815560766638, −10.62079957473011977210056735060, −9.613485605778233409037572493878, −8.109692422308539715768167406007, −7.51075944296788459771986792092, −6.53640129464565779297812486393, −5.37907321798644365385093673272, −4.06924957543985102308657984921, −2.95685633149614110841915955503, −1.59292592098147301756862829467, 1.00120598285003837238385451337, 1.64767391668824349973636454379, 3.87727173175322673923655948256, 5.10112831128849623913593874991, 5.32602105978001685757609516273, 7.22238265861442759432356806512, 8.131735542502964671303403842882, 8.696350991475427930915421249943, 9.678504353708465524666066049825, 10.90532095902347255940642236479

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