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Properties

Label 2-153-153.106-c1-0-4
Degree $2$
Conductor $153$
Sign $-0.166 - 0.986i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.25 + 1.30i)2-s + (−1.36 + 1.06i)3-s + (2.39 + 4.14i)4-s + (−3.16 − 0.847i)5-s + (−4.47 + 0.619i)6-s + (3.71 − 0.996i)7-s + 7.26i·8-s + (0.737 − 2.90i)9-s + (−6.03 − 6.03i)10-s + (0.362 − 0.0971i)11-s + (−7.68 − 3.12i)12-s + (1.09 + 1.88i)13-s + (9.69 + 2.59i)14-s + (5.22 − 2.20i)15-s + (−4.68 + 8.10i)16-s + (−1.79 − 3.71i)17-s + ⋯
L(s)  = 1  + (1.59 + 0.921i)2-s + (−0.789 + 0.614i)3-s + (1.19 + 2.07i)4-s + (−1.41 − 0.378i)5-s + (−1.82 + 0.252i)6-s + (1.40 − 0.376i)7-s + 2.56i·8-s + (0.245 − 0.969i)9-s + (−1.90 − 1.90i)10-s + (0.109 − 0.0292i)11-s + (−2.21 − 0.901i)12-s + (0.302 + 0.524i)13-s + (2.59 + 0.694i)14-s + (1.34 − 0.569i)15-s + (−1.17 + 2.02i)16-s + (−0.435 − 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.166 - 0.986i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -0.166 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20181 + 1.42192i\)
\(L(\frac12)\) \(\approx\) \(1.20181 + 1.42192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.36 - 1.06i)T \)
17 \( 1 + (1.79 + 3.71i)T \)
good2 \( 1 + (-2.25 - 1.30i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (3.16 + 0.847i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.71 + 0.996i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.362 + 0.0971i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.09 - 1.88i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 0.425iT - 19T^{2} \)
23 \( 1 + (-0.956 + 3.57i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.0646 + 0.241i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (6.87 + 1.84i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.47 - 3.47i)T - 37iT^{2} \)
41 \( 1 + (1.28 - 4.80i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.98 + 4.03i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.831 + 1.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.95iT - 53T^{2} \)
59 \( 1 + (1.53 - 0.886i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.61 + 1.50i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.487 + 0.843i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.74 - 8.74i)T - 71iT^{2} \)
73 \( 1 + (-2.29 + 2.29i)T - 73iT^{2} \)
79 \( 1 + (-9.75 + 2.61i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.51 + 2.03i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 14.9T + 89T^{2} \)
97 \( 1 + (-4.46 - 16.6i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.35836428718281674770350884117, −12.10190514097234207193668975387, −11.62445021570982794668009947855, −10.95677581004522230724835878331, −8.683545652023105794978755428941, −7.61584020671661694960181177430, −6.70127542100710732964204308147, −5.14134573862976817152920405903, −4.55866061585288129656950209744, −3.73589100939933753138588332426, 1.77230610875665996407745568796, 3.63278505650394798085468987193, 4.77440388757985805651226434570, 5.71656080857390187545255630613, 7.10358237761565801621502802957, 8.212138381670436034675803080510, 10.69196490376267239215740283379, 11.10571330205548880661823801273, 11.80579901892356961239176832057, 12.44510934533974745163621044365

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