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Properties

Label 2-153-17.13-c3-0-4
Degree $2$
Conductor $153$
Sign $-0.869 + 0.493i$
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  + 4.08i·2-s − 8.65·4-s + (5.03 + 5.03i)5-s + (−6.78 + 6.78i)7-s − 2.67i·8-s + (−20.5 + 20.5i)10-s + (−16.3 + 16.3i)11-s − 9.56·13-s + (−27.6 − 27.6i)14-s − 58.3·16-s + (−26.7 + 64.7i)17-s − 81.7i·19-s + (−43.5 − 43.5i)20-s + (−66.5 − 66.5i)22-s + (−61.2 + 61.2i)23-s + ⋯
L(s)  = 1  + 1.44i·2-s − 1.08·4-s + (0.450 + 0.450i)5-s + (−0.366 + 0.366i)7-s − 0.118i·8-s + (−0.649 + 0.649i)10-s + (−0.447 + 0.447i)11-s − 0.204·13-s + (−0.528 − 0.528i)14-s − 0.911·16-s + (−0.381 + 0.924i)17-s − 0.986i·19-s + (−0.487 − 0.487i)20-s + (−0.645 − 0.645i)22-s + (−0.554 + 0.554i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.869 + 0.493i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.869 + 0.493i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.314084 - 1.18969i\)
\(L(\frac12)\) \(\approx\) \(0.314084 - 1.18969i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (26.7 - 64.7i)T \)
good2 \( 1 - 4.08iT - 8T^{2} \)
5 \( 1 + (-5.03 - 5.03i)T + 125iT^{2} \)
7 \( 1 + (6.78 - 6.78i)T - 343iT^{2} \)
11 \( 1 + (16.3 - 16.3i)T - 1.33e3iT^{2} \)
13 \( 1 + 9.56T + 2.19e3T^{2} \)
19 \( 1 + 81.7iT - 6.85e3T^{2} \)
23 \( 1 + (61.2 - 61.2i)T - 1.21e4iT^{2} \)
29 \( 1 + (19.3 + 19.3i)T + 2.43e4iT^{2} \)
31 \( 1 + (-149. - 149. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-167. - 167. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-251. + 251. i)T - 6.89e4iT^{2} \)
43 \( 1 + 63.1iT - 7.95e4T^{2} \)
47 \( 1 + 382.T + 1.03e5T^{2} \)
53 \( 1 - 685. iT - 1.48e5T^{2} \)
59 \( 1 - 263. iT - 2.05e5T^{2} \)
61 \( 1 + (374. - 374. i)T - 2.26e5iT^{2} \)
67 \( 1 - 423.T + 3.00e5T^{2} \)
71 \( 1 + (97.6 + 97.6i)T + 3.57e5iT^{2} \)
73 \( 1 + (-713. - 713. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-433. + 433. i)T - 4.93e5iT^{2} \)
83 \( 1 - 1.12e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.59e3T + 7.04e5T^{2} \)
97 \( 1 + (1.08e3 + 1.08e3i)T + 9.12e5iT^{2} \)
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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

−13.45410507746157609790464677660, −12.33302073975970659359546443336, −10.93869685328477200594143468883, −9.797493636142955738709795887757, −8.723638084044689232764015701978, −7.65483081950555907410406524718, −6.60930210165670133153373356521, −5.85639408296167214160382070434, −4.58737136383775707386361359706, −2.50748867666942829837509510535, 0.56901546269346086813234228614, 2.16650879327873395545793989912, 3.50369992154376163095427293755, 4.85224668455351300463003078319, 6.36520118884272721958731830099, 7.955399121228946781283524837226, 9.366190609874013774937937996364, 9.943140378156687520695472528050, 10.98448596697421595537593531321, 11.83927781442131933424612423062

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