↑

Properties

Label 2-152-152.67-c3-0-11
Degree $2$
Conductor $152$
Sign $-0.657 - 0.753i$
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  + (−2.78 + 0.491i)2-s + (6.01 + 7.16i)3-s + (7.51 − 2.73i)4-s + (−20.2 − 17.0i)6-s + (−19.5 + 11.3i)8-s + (−10.5 + 59.6i)9-s + (15.8 + 27.4i)11-s + (64.8 + 37.4i)12-s + (49.0 − 41.1i)16-s + (8.34 + 47.3i)17-s − 171. i·18-s + (−71.5 − 41.6i)19-s + (−57.5 − 68.6i)22-s + (−198. − 72.4i)24-s + (95.7 + 80.3i)25-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (1.15 + 1.37i)3-s + (0.939 − 0.342i)4-s + (−1.37 − 1.15i)6-s + (−0.866 + 0.500i)8-s + (−0.389 + 2.20i)9-s + (0.433 + 0.751i)11-s + (1.55 + 0.900i)12-s + (0.766 − 0.642i)16-s + (0.119 + 0.675i)17-s − 2.24i·18-s + (−0.864 − 0.503i)19-s + (−0.557 − 0.664i)22-s + (−1.69 − 0.615i)24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.657 - 0.753i$
Analytic conductor: \(8.96829\)
Root analytic conductor: \(2.99471\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :3/2),\ -0.657 - 0.753i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.608313 + 1.33839i\)
\(L(\frac12)\) \(\approx\) \(0.608313 + 1.33839i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.78 - 0.491i)T \)
19 \( 1 + (71.5 + 41.6i)T \)
good3 \( 1 + (-6.01 - 7.16i)T + (-4.68 + 26.5i)T^{2} \)
5 \( 1 + (-95.7 - 80.3i)T^{2} \)
7 \( 1 + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-15.8 - 27.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-8.34 - 47.3i)T + (-4.61e3 + 1.68e3i)T^{2} \)
23 \( 1 + (-9.32e3 + 7.82e3i)T^{2} \)
29 \( 1 + (-2.29e4 - 8.34e3i)T^{2} \)
31 \( 1 + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 5.06e4T^{2} \)
41 \( 1 + (-148. - 177. i)T + (-1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (529. + 192. i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + (1.14e5 - 9.56e4i)T^{2} \)
59 \( 1 + (-780. + 137. i)T + (1.92e5 - 7.02e4i)T^{2} \)
61 \( 1 + (-1.73e5 + 1.45e5i)T^{2} \)
67 \( 1 + (-989. - 174. i)T + (2.82e5 + 1.02e5i)T^{2} \)
71 \( 1 + (2.74e5 + 2.30e5i)T^{2} \)
73 \( 1 + (-828. + 695. i)T + (6.75e4 - 3.83e5i)T^{2} \)
79 \( 1 + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (750. - 1.30e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-998. + 1.18e3i)T + (-1.22e5 - 6.94e5i)T^{2} \)
97 \( 1 + (-690. + 121. i)T + (8.57e5 - 3.12e5i)T^{2} \)
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

−12.96223477658494971783905144126, −11.39979544961405474650181398290, −10.41011045896599701014566174721, −9.728457385777116695349208429615, −8.852521234056794221984483123096, −8.138242389345972128736139124804, −6.78789371008576191925981448444, −4.99652956535270464342227418017, −3.57859315060834975182625151706, −2.13492000025554616971096775820, 0.856748576121686104493027374618, 2.22075539093337598479561376346, 3.39619935824342537616619862505, 6.27447333148520681368330301809, 7.08377941808247979339685443593, 8.198123186365362086699420802683, 8.700661126753645737556492718148, 9.793807277229731576281343492364, 11.22892870130606601609593593685, 12.22734625895099001876013048595

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