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Properties

Label 2-153-153.106-c1-0-15
Degree $2$
Conductor $153$
Sign $-0.928 + 0.372i$
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  + (−0.268 − 0.155i)2-s + (−0.546 − 1.64i)3-s + (−0.951 − 1.64i)4-s + (−0.355 − 0.0951i)5-s + (−0.108 + 0.526i)6-s + (−3.67 + 0.984i)7-s + 1.21i·8-s + (−2.40 + 1.79i)9-s + (0.0807 + 0.0807i)10-s + (3.38 − 0.906i)11-s + (−2.18 + 2.46i)12-s + (−1.06 − 1.84i)13-s + (1.14 + 0.305i)14-s + (0.0376 + 0.635i)15-s + (−1.71 + 2.97i)16-s + (0.919 − 4.01i)17-s + ⋯
L(s)  = 1  + (−0.190 − 0.109i)2-s + (−0.315 − 0.948i)3-s + (−0.475 − 0.824i)4-s + (−0.158 − 0.0425i)5-s + (−0.0442 + 0.215i)6-s + (−1.38 + 0.371i)7-s + 0.428i·8-s + (−0.801 + 0.598i)9-s + (0.0255 + 0.0255i)10-s + (1.01 − 0.273i)11-s + (−0.632 + 0.711i)12-s + (−0.295 − 0.512i)13-s + (0.304 + 0.0816i)14-s + (0.00971 + 0.164i)15-s + (−0.428 + 0.742i)16-s + (0.222 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.928 + 0.372i$
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: \((2,\ 153,\ (\ :1/2),\ -0.928 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103827 - 0.537389i\)
\(L(\frac12)\) \(\approx\) \(0.103827 - 0.537389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.546 + 1.64i)T \)
17 \( 1 + (-0.919 + 4.01i)T \)
good2 \( 1 + (0.268 + 0.155i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.355 + 0.0951i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.67 - 0.984i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.38 + 0.906i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.06 + 1.84i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 6.27iT - 19T^{2} \)
23 \( 1 + (-1.95 + 7.27i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.62 - 6.05i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.58 - 0.960i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.55 - 1.55i)T - 37iT^{2} \)
41 \( 1 + (1.46 - 5.45i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.10 + 1.21i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.72 + 4.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.5iT - 53T^{2} \)
59 \( 1 + (-1.45 + 0.841i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.6 + 2.86i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.09 + 8.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.69 + 5.69i)T - 71iT^{2} \)
73 \( 1 + (-0.753 + 0.753i)T - 73iT^{2} \)
79 \( 1 + (1.48 - 0.398i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (6.88 + 3.97i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + (-0.810 - 3.02i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.55077606886334003395121836576, −11.66981246523545512214461318751, −10.50035267656766390863960107170, −9.381912331228471666194096694284, −8.590208204408068804712688085840, −6.89306354042210267665688788748, −6.22915295808188933719396683276, −4.94131984908621976385785182429, −2.79538519858926423442530104205, −0.59626680391382347790108491376, 3.58706815513553908128994883878, 4.01863326924608318833120603534, 5.88231003694345318067663754746, 7.05486523971490413623853723294, 8.427637355568872137511945348328, 9.699382808291933603469316389003, 9.850099604548104818977991521666, 11.56468001312814622150950327398, 12.28304697024726628538122644540, 13.35299633595707930539164108138

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