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Properties

Label 2-154-7.2-c3-0-18
Degree $2$
Conductor $154$
Sign $-0.918 - 0.395i$
Analytic cond. $9.08629$
Root an. cond. $3.01434$
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 − 1.73i)2-s + (−2.24 − 3.89i)3-s + (−1.99 − 3.46i)4-s + (5.50 − 9.53i)5-s − 8.98·6-s + (−18.4 + 1.00i)7-s − 7.99·8-s + (3.40 − 5.89i)9-s + (−11.0 − 19.0i)10-s + (5.5 + 9.52i)11-s + (−8.98 + 15.5i)12-s − 24.3·13-s + (−16.7 + 33.0i)14-s − 49.4·15-s + (−8 + 13.8i)16-s + (5.17 + 8.96i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.432 − 0.748i)3-s + (−0.249 − 0.433i)4-s + (0.492 − 0.852i)5-s − 0.611·6-s + (−0.998 + 0.0540i)7-s − 0.353·8-s + (0.126 − 0.218i)9-s + (−0.348 − 0.602i)10-s + (0.150 + 0.261i)11-s + (−0.216 + 0.374i)12-s − 0.520·13-s + (−0.319 + 0.630i)14-s − 0.851·15-s + (−0.125 + 0.216i)16-s + (0.0738 + 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(9.08629\)
Root analytic conductor: \(3.01434\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :3/2),\ -0.918 - 0.395i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.228009 + 1.10718i\)
\(L(\frac12)\) \(\approx\) \(0.228009 + 1.10718i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
7 \( 1 + (18.4 - 1.00i)T \)
11 \( 1 + (-5.5 - 9.52i)T \)
good3 \( 1 + (2.24 + 3.89i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.50 + 9.53i)T + (-62.5 - 108. i)T^{2} \)
13 \( 1 + 24.3T + 2.19e3T^{2} \)
17 \( 1 + (-5.17 - 8.96i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-12.1 + 21.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (46.1 - 79.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 98.1T + 2.43e4T^{2} \)
31 \( 1 + (131. + 227. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-108. + 188. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 304.T + 6.89e4T^{2} \)
43 \( 1 - 22.5T + 7.95e4T^{2} \)
47 \( 1 + (-77.2 + 133. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (295. + 511. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (126. + 219. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-401. + 695. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (152. + 264. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 151.T + 3.57e5T^{2} \)
73 \( 1 + (190. + 329. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (455. - 789. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.17e3T + 5.71e5T^{2} \)
89 \( 1 + (-271. + 470. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 601.T + 9.12e5T^{2} \)
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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

−12.23737905280578553570130192186, −11.20955690947041687508013057777, −9.656462800685400722297498679363, −9.331667484671245760107924236987, −7.51294895852707600105262384923, −6.28542061971824235182089445122, −5.36435528938917658082722413189, −3.80187823782013589958025241467, −1.98417361697996503647217810979, −0.49431530373006972865436542538, 2.84887178683083539520168924372, 4.21593794753903874807638667769, 5.57392744124917705366604165490, 6.48149549736111806778519359708, 7.51553160221845447025200360553, 9.156647428421606334938717614142, 10.10535800568836981763797751058, 10.80460691258233071814495916945, 12.17572441257995565988115547462, 13.21269617713243176886312303672

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