Properties

Label 2-153-153.106-c3-0-13
Degree $2$
Conductor $153$
Sign $0.335 - 0.941i$
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  + (0.596 + 0.344i)2-s + (−5.19 + 0.226i)3-s + (−3.76 − 6.51i)4-s + (9.82 + 2.63i)5-s + (−3.17 − 1.65i)6-s + (−27.3 + 7.32i)7-s − 10.6i·8-s + (26.8 − 2.34i)9-s + (4.95 + 4.95i)10-s + (15.0 − 4.02i)11-s + (21.0 + 32.9i)12-s + (27.3 + 47.3i)13-s + (−18.8 − 5.04i)14-s + (−51.5 − 11.4i)15-s + (−26.4 + 45.7i)16-s + (64.7 + 26.9i)17-s + ⋯
L(s)  = 1  + (0.210 + 0.121i)2-s + (−0.999 + 0.0434i)3-s + (−0.470 − 0.814i)4-s + (0.878 + 0.235i)5-s + (−0.215 − 0.112i)6-s + (−1.47 + 0.395i)7-s − 0.472i·8-s + (0.996 − 0.0869i)9-s + (0.156 + 0.156i)10-s + (0.411 − 0.110i)11-s + (0.505 + 0.793i)12-s + (0.583 + 1.01i)13-s + (−0.359 − 0.0962i)14-s + (−0.887 − 0.196i)15-s + (−0.412 + 0.715i)16-s + (0.923 + 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.335 - 0.941i$
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: \((2,\ 153,\ (\ :3/2),\ 0.335 - 0.941i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.821017 + 0.579025i\)
\(L(\frac12)\) \(\approx\) \(0.821017 + 0.579025i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.19 - 0.226i)T \)
17 \( 1 + (-64.7 - 26.9i)T \)
good2 \( 1 + (-0.596 - 0.344i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-9.82 - 2.63i)T + (108. + 62.5i)T^{2} \)
7 \( 1 + (27.3 - 7.32i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (-15.0 + 4.02i)T + (1.15e3 - 665.5i)T^{2} \)
13 \( 1 + (-27.3 - 47.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
19 \( 1 - 60.2iT - 6.85e3T^{2} \)
23 \( 1 + (17.6 - 65.7i)T + (-1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (-73.4 - 274. i)T + (-2.11e4 + 1.21e4i)T^{2} \)
31 \( 1 + (-258. - 69.3i)T + (2.57e4 + 1.48e4i)T^{2} \)
37 \( 1 + (-79.8 + 79.8i)T - 5.06e4iT^{2} \)
41 \( 1 + (-31.4 + 117. i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (426. + 246. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (83.8 - 145. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 27.0iT - 1.48e5T^{2} \)
59 \( 1 + (-260. + 150. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (186. - 49.9i)T + (1.96e5 - 1.13e5i)T^{2} \)
67 \( 1 + (-193. - 335. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (215. - 215. i)T - 3.57e5iT^{2} \)
73 \( 1 + (244. - 244. i)T - 3.89e5iT^{2} \)
79 \( 1 + (-568. + 152. i)T + (4.26e5 - 2.46e5i)T^{2} \)
83 \( 1 + (607. + 350. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 785.T + 7.04e5T^{2} \)
97 \( 1 + (-435. - 1.62e3i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−12.75999362329770726277673237158, −11.81220787684807184427189834140, −10.33963809111659823198848765318, −9.924269755213406434266612282633, −8.998386498087324507040179381003, −6.66091880135120304556809389657, −6.21197456903826411946977314582, −5.34209714995964032036974284658, −3.74468281954541697453897088613, −1.38924186766287025093965286438, 0.57075757697044343019937330084, 3.05292989803463939445106100325, 4.45531521921525835414670655444, 5.81050464289249635609717763622, 6.66254043193003021416807195212, 8.067947804587622478410692421393, 9.658526325199099808522000453918, 10.03400768953020566560058268752, 11.55450454880046120331038542480, 12.44573057185738271698087160937

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