Properties

Label 2-153-153.106-c1-0-14
Degree $2$
Conductor $153$
Sign $0.209 + 0.977i$
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  + (0.218 + 0.126i)2-s + (1.43 − 0.963i)3-s + (−0.968 − 1.67i)4-s + (−3.83 − 1.02i)5-s + (0.436 − 0.0289i)6-s + (2.05 − 0.551i)7-s − 0.995i·8-s + (1.14 − 2.77i)9-s + (−0.709 − 0.709i)10-s + (1.14 − 0.307i)11-s + (−3.00 − 1.48i)12-s + (3.09 + 5.36i)13-s + (0.520 + 0.139i)14-s + (−6.50 + 2.21i)15-s + (−1.81 + 3.13i)16-s + (2.45 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.154 + 0.0893i)2-s + (0.831 − 0.556i)3-s + (−0.484 − 0.838i)4-s + (−1.71 − 0.459i)5-s + (0.178 − 0.0118i)6-s + (0.778 − 0.208i)7-s − 0.351i·8-s + (0.381 − 0.924i)9-s + (−0.224 − 0.224i)10-s + (0.346 − 0.0928i)11-s + (−0.868 − 0.427i)12-s + (0.858 + 1.48i)13-s + (0.139 + 0.0372i)14-s + (−1.67 + 0.571i)15-s + (−0.452 + 0.783i)16-s + (0.595 + 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.209 + 0.977i$
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: \((2,\ 153,\ (\ :1/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929182 - 0.751409i\)
\(L(\frac12)\) \(\approx\) \(0.929182 - 0.751409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.43 + 0.963i)T \)
17 \( 1 + (-2.45 - 3.31i)T \)
good2 \( 1 + (-0.218 - 0.126i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (3.83 + 1.02i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.05 + 0.551i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.14 + 0.307i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.09 - 5.36i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 5.66iT - 19T^{2} \)
23 \( 1 + (0.188 - 0.703i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.799 + 2.98i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.0434 - 0.0116i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.68 + 1.68i)T - 37iT^{2} \)
41 \( 1 + (0.174 - 0.650i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.442 + 0.255i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.10iT - 53T^{2} \)
59 \( 1 + (-7.91 + 4.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.59 - 1.49i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.68 + 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.32 + 2.32i)T - 71iT^{2} \)
73 \( 1 + (8.24 - 8.24i)T - 73iT^{2} \)
79 \( 1 + (0.124 - 0.0332i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-5.37 - 3.10i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.09T + 89T^{2} \)
97 \( 1 + (-1.05 - 3.94i)T + (-84.0 + 48.5i)T^{2} \)
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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

−12.90003360734374423415905061616, −11.74014084822107979600818680433, −11.04324603686401138776499806286, −9.286351579106586510270682265625, −8.578779764883221528577926299802, −7.66194786286520706863413636763, −6.48421656723646199930048237884, −4.56202474075457948304831939025, −3.85108895679071928163403863701, −1.27117305772406437416122209008, 3.19509858292154295696005290862, 3.80353997092318172205848665558, 5.02083606964130983430330742324, 7.46934631412147756869816482624, 8.079774179138564966296218966703, 8.637571502158913748211082393685, 10.24389589889740821990669357354, 11.34516242698984307290818386292, 12.11228083061361717532731434923, 13.17993136915825202484207020851

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