Properties

Label 2-153-153.106-c3-0-25
Degree $2$
Conductor $153$
Sign $-0.0104 + 0.999i$
Analytic cond. $9.02729$
Root an. cond. $3.00454$
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  + (−1.40 − 0.811i)2-s + (−2.42 + 4.59i)3-s + (−2.68 − 4.64i)4-s + (7.52 + 2.01i)5-s + (7.13 − 4.49i)6-s + (−6.20 + 1.66i)7-s + 21.6i·8-s + (−15.2 − 22.2i)9-s + (−8.93 − 8.93i)10-s + (−11.2 + 3.00i)11-s + (27.8 − 1.08i)12-s + (10.5 + 18.3i)13-s + (10.0 + 2.70i)14-s + (−27.4 + 29.7i)15-s + (−3.85 + 6.68i)16-s + (38.1 − 58.7i)17-s + ⋯
L(s)  = 1  + (−0.496 − 0.286i)2-s + (−0.466 + 0.884i)3-s + (−0.335 − 0.580i)4-s + (0.672 + 0.180i)5-s + (0.485 − 0.305i)6-s + (−0.335 + 0.0898i)7-s + 0.958i·8-s + (−0.565 − 0.824i)9-s + (−0.282 − 0.282i)10-s + (−0.307 + 0.0823i)11-s + (0.670 − 0.0259i)12-s + (0.225 + 0.391i)13-s + (0.192 + 0.0515i)14-s + (−0.473 + 0.511i)15-s + (−0.0602 + 0.104i)16-s + (0.544 − 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0104 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0104 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.0104 + 0.999i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.0104 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.491286 - 0.496425i\)
\(L(\frac12)\) \(\approx\) \(0.491286 - 0.496425i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.42 - 4.59i)T \)
17 \( 1 + (-38.1 + 58.7i)T \)
good2 \( 1 + (1.40 + 0.811i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-7.52 - 2.01i)T + (108. + 62.5i)T^{2} \)
7 \( 1 + (6.20 - 1.66i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (11.2 - 3.00i)T + (1.15e3 - 665.5i)T^{2} \)
13 \( 1 + (-10.5 - 18.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
19 \( 1 + 153. iT - 6.85e3T^{2} \)
23 \( 1 + (-18.3 + 68.3i)T + (-1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (76.9 + 287. i)T + (-2.11e4 + 1.21e4i)T^{2} \)
31 \( 1 + (-220. - 59.1i)T + (2.57e4 + 1.48e4i)T^{2} \)
37 \( 1 + (235. - 235. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-124. + 465. i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (34.8 + 20.1i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (72.5 - 125. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 216. iT - 1.48e5T^{2} \)
59 \( 1 + (-127. + 73.8i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-27.1 + 7.28i)T + (1.96e5 - 1.13e5i)T^{2} \)
67 \( 1 + (-406. - 703. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (429. - 429. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-600. + 600. i)T - 3.89e5iT^{2} \)
79 \( 1 + (608. - 163. i)T + (4.26e5 - 2.46e5i)T^{2} \)
83 \( 1 + (-242. - 139. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 68.8T + 7.04e5T^{2} \)
97 \( 1 + (229. + 858. i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−11.85426943795024495283748784367, −11.02056895945145737433845141869, −10.01344554430596871649428515316, −9.558800319325617132455983530146, −8.575411176708139285821799414411, −6.62919951450656838414998233727, −5.54825800752127026814363616909, −4.55935237555977938611722470740, −2.60143952088147818044149063439, −0.44654294279438890784725852584, 1.42848915380552162100486178296, 3.45418284207542107056716588896, 5.42871386184297325367915314706, 6.40129857709179952787687513091, 7.66117202551791353392483642372, 8.338354264749195243233118839228, 9.643768786464068986109309587029, 10.59561866090206492789700643796, 12.07270299156573288504637928041, 12.80690650729982476618316576684

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