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Properties

Label 2-153-153.106-c1-0-6
Degree $2$
Conductor $153$
Sign $0.412 - 0.910i$
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  + (1.84 + 1.06i)2-s + (−1.73 − 0.00129i)3-s + (1.27 + 2.20i)4-s + (3.52 + 0.945i)5-s + (−3.19 − 1.84i)6-s + (−3.69 + 0.990i)7-s + 1.15i·8-s + (2.99 + 0.00449i)9-s + (5.50 + 5.50i)10-s + (0.437 − 0.117i)11-s + (−2.19 − 3.81i)12-s + (−0.231 − 0.401i)13-s + (−7.88 − 2.11i)14-s + (−6.11 − 1.64i)15-s + (1.31 − 2.26i)16-s + (−3.81 − 1.56i)17-s + ⋯
L(s)  = 1  + (1.30 + 0.753i)2-s + (−0.999 − 0.000748i)3-s + (0.635 + 1.10i)4-s + (1.57 + 0.422i)5-s + (−1.30 − 0.754i)6-s + (−1.39 + 0.374i)7-s + 0.408i·8-s + (0.999 + 0.00149i)9-s + (1.74 + 1.74i)10-s + (0.131 − 0.0353i)11-s + (−0.634 − 1.10i)12-s + (−0.0642 − 0.111i)13-s + (−2.10 − 0.564i)14-s + (−1.57 − 0.424i)15-s + (0.327 − 0.567i)16-s + (−0.925 − 0.378i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.412 - 0.910i$
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: \((2,\ 153,\ (\ :1/2),\ 0.412 - 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48295 + 0.956376i\)
\(L(\frac12)\) \(\approx\) \(1.48295 + 0.956376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.00129i)T \)
17 \( 1 + (3.81 + 1.56i)T \)
good2 \( 1 + (-1.84 - 1.06i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-3.52 - 0.945i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.69 - 0.990i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.437 + 0.117i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.231 + 0.401i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 4.69iT - 19T^{2} \)
23 \( 1 + (0.317 - 1.18i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.13 + 7.95i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.26 - 0.875i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.98 - 3.98i)T - 37iT^{2} \)
41 \( 1 + (2.47 - 9.24i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.86 - 2.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.56 - 2.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.65iT - 53T^{2} \)
59 \( 1 + (6.02 - 3.47i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.19 - 1.66i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-6.26 - 10.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.73 + 1.73i)T - 71iT^{2} \)
73 \( 1 + (4.28 - 4.28i)T - 73iT^{2} \)
79 \( 1 + (-11.3 + 3.05i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.377 + 0.218i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 + (1.90 + 7.09i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.32941506659395299342594199099, −12.63286909142334479109812618677, −11.42677934380044738163274626556, −10.03768250674551349169428822606, −9.434532138863566741010561258437, −6.98889442362273358305997785875, −6.33562414925598469455806793110, −5.80781251555879395799976789136, −4.63244210493018598623930661926, −2.81898831160972205086696201267, 1.90447388520619164191513514785, 3.72186642425258321558523098341, 5.08482002746423674195626222187, 5.98890869031783399570426074336, 6.67594642427589717915446361840, 9.181585179054516956413883043751, 10.21879223467155007389022056721, 10.77284827311997008006302990765, 12.31791130280921744660632208130, 12.69463899876240348792252707562

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