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Properties

Label 2-152-19.7-c3-0-14
Degree $2$
Conductor $152$
Sign $-0.517 - 0.855i$
Analytic cond. $8.96829$
Root an. cond. $2.99471$
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  + (−3.20 − 5.54i)3-s + (−9.34 − 16.1i)5-s − 16.3·7-s + (−7.01 + 12.1i)9-s + 53.5·11-s + (−42.4 + 73.5i)13-s + (−59.8 + 103. i)15-s + (−24.0 − 41.5i)17-s + (80.5 − 19.1i)19-s + (52.2 + 90.5i)21-s + (55.9 − 96.8i)23-s + (−111. + 193. i)25-s − 83.0·27-s + (−131. + 227. i)29-s − 122.·31-s + ⋯
L(s)  = 1  + (−0.616 − 1.06i)3-s + (−0.835 − 1.44i)5-s − 0.880·7-s + (−0.259 + 0.450i)9-s + 1.46·11-s + (−0.906 + 1.56i)13-s + (−1.02 + 1.78i)15-s + (−0.342 − 0.593i)17-s + (0.972 − 0.231i)19-s + (0.543 + 0.940i)21-s + (0.507 − 0.878i)23-s + (−0.895 + 1.55i)25-s − 0.592·27-s + (−0.840 + 1.45i)29-s − 0.708·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.517 - 0.855i$
Analytic conductor: \(8.96829\)
Root analytic conductor: \(2.99471\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :3/2),\ -0.517 - 0.855i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.156566 + 0.277661i\)
\(L(\frac12)\) \(\approx\) \(0.156566 + 0.277661i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-80.5 + 19.1i)T \)
good3 \( 1 + (3.20 + 5.54i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (9.34 + 16.1i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + 16.3T + 343T^{2} \)
11 \( 1 - 53.5T + 1.33e3T^{2} \)
13 \( 1 + (42.4 - 73.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (24.0 + 41.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
23 \( 1 + (-55.9 + 96.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (131. - 227. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 122.T + 2.97e4T^{2} \)
37 \( 1 + 32.6T + 5.06e4T^{2} \)
41 \( 1 + (207. + 359. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (29.2 + 50.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (85.2 - 147. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-165. + 287. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-354. - 613. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-8.95 + 15.5i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-164. + 285. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (31.4 + 54.4i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (281. + 487. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (154. + 266. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 524.T + 5.71e5T^{2} \)
89 \( 1 + (298. - 516. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-275. - 476. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.04036003686982510776238652752, −11.46218467394019592195789922122, −9.349187577792674429871427615361, −8.946235227844610732055059506529, −7.20391748210400753446565413352, −6.75903162616693257444396516950, −5.16442679385987294131804086684, −3.92269903628279132438640292823, −1.46476850908603937665224022157, −0.17378942818915227529490669437, 3.18344663352096070934216950970, 3.94582057062797977588807342984, 5.59292085709248664915037726079, 6.75100040057492582350802773521, 7.76879661407297267042342294634, 9.622434762248073582922705288065, 10.11324950667356859672770026863, 11.19949511392723976146561322374, 11.75408858585301542791026341997, 13.09815901937935005275268198774

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