↑

Properties

Label 2-152-19.11-c3-0-10
Degree $2$
Conductor $152$
Sign $0.505 + 0.862i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.975 − 1.68i)3-s + (4.98 − 8.63i)5-s + 13.4·7-s + (11.5 + 20.0i)9-s − 4.68·11-s + (−35.2 − 61.0i)13-s + (−9.71 − 16.8i)15-s + (36.3 − 62.9i)17-s + (55.2 − 61.7i)19-s + (13.1 − 22.7i)21-s + (24.5 + 42.5i)23-s + (12.8 + 22.2i)25-s + 97.9·27-s + (−93.0 − 161. i)29-s + 147.·31-s + ⋯
L(s)  = 1  + (0.187 − 0.325i)3-s + (0.445 − 0.771i)5-s + 0.727·7-s + (0.429 + 0.743i)9-s − 0.128·11-s + (−0.752 − 1.30i)13-s + (−0.167 − 0.289i)15-s + (0.518 − 0.898i)17-s + (0.666 − 0.745i)19-s + (0.136 − 0.236i)21-s + (0.222 + 0.385i)23-s + (0.102 + 0.177i)25-s + 0.697·27-s + (−0.595 − 1.03i)29-s + 0.856·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.505 + 0.862i$
Analytic conductor: \(8.96829\)
Root analytic conductor: \(2.99471\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :3/2),\ 0.505 + 0.862i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76277 - 1.00973i\)
\(L(\frac12)\) \(\approx\) \(1.76277 - 1.00973i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-55.2 + 61.7i)T \)
good3 \( 1 + (-0.975 + 1.68i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-4.98 + 8.63i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 13.4T + 343T^{2} \)
11 \( 1 + 4.68T + 1.33e3T^{2} \)
13 \( 1 + (35.2 + 61.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-36.3 + 62.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-24.5 - 42.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (93.0 + 161. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 + 209.T + 5.06e4T^{2} \)
41 \( 1 + (-173. + 299. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (203. - 352. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-220. - 382. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (14.2 + 24.7i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (317. - 549. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (5.75 + 9.96i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-292. - 507. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-167. + 289. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (370. - 642. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (47.3 - 82.0i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 313.T + 5.71e5T^{2} \)
89 \( 1 + (-749. - 1.29e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (180. - 312. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.53514263125753105730673746787, −11.42719262955137574237673241035, −10.24227898915857478707477218743, −9.274864461642859038150840546334, −7.993147160229946429280759102691, −7.36283486541124657334075834218, −5.45410961965375522538895832591, −4.80652313207559081509191882402, −2.65946255756573947867835360600, −1.08967050245649309167860179229, 1.78455935377545302535057507333, 3.46647544784674947994234772736, 4.81717435441929227044954275082, 6.31968576780463731647007659800, 7.29709216658506216294108126268, 8.649325189063301124448894999634, 9.778985454842602167777118417376, 10.49008417442220965866273596538, 11.69939688426451200226366659687, 12.54959603051103061650587756664

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