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Properties

Label 2-153-17.9-c3-0-7
Degree $2$
Conductor $153$
Sign $-0.256 - 0.966i$
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  + (−0.320 + 0.320i)2-s + 7.79i·4-s + (2.81 − 6.79i)5-s + (6.62 + 15.9i)7-s + (−5.05 − 5.05i)8-s + (1.27 + 3.07i)10-s + (0.780 − 0.323i)11-s + 29.3i·13-s + (−7.23 − 2.99i)14-s − 59.1·16-s + (14.8 + 68.5i)17-s + (−13.1 + 13.1i)19-s + (52.9 + 21.9i)20-s + (−0.146 + 0.353i)22-s + (−111. + 46.0i)23-s + ⋯
L(s)  = 1  + (−0.113 + 0.113i)2-s + 0.974i·4-s + (0.251 − 0.607i)5-s + (0.357 + 0.863i)7-s + (−0.223 − 0.223i)8-s + (0.0403 + 0.0972i)10-s + (0.0213 − 0.00885i)11-s + 0.626i·13-s + (−0.138 − 0.0572i)14-s − 0.923·16-s + (0.211 + 0.977i)17-s + (−0.158 + 0.158i)19-s + (0.592 + 0.245i)20-s + (−0.00141 + 0.00342i)22-s + (−1.00 + 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ -0.256 - 0.966i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.869812 + 1.13127i\)
\(L(\frac12)\) \(\approx\) \(0.869812 + 1.13127i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14.8 - 68.5i)T \)
good2 \( 1 + (0.320 - 0.320i)T - 8iT^{2} \)
5 \( 1 + (-2.81 + 6.79i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (-6.62 - 15.9i)T + (-242. + 242. i)T^{2} \)
11 \( 1 + (-0.780 + 0.323i)T + (941. - 941. i)T^{2} \)
13 \( 1 - 29.3iT - 2.19e3T^{2} \)
19 \( 1 + (13.1 - 13.1i)T - 6.85e3iT^{2} \)
23 \( 1 + (111. - 46.0i)T + (8.60e3 - 8.60e3i)T^{2} \)
29 \( 1 + (76.9 - 185. i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + (-98.5 - 40.8i)T + (2.10e4 + 2.10e4i)T^{2} \)
37 \( 1 + (64.8 + 26.8i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (73.4 + 177. i)T + (-4.87e4 + 4.87e4i)T^{2} \)
43 \( 1 + (-359. - 359. i)T + 7.95e4iT^{2} \)
47 \( 1 + 235. iT - 1.03e5T^{2} \)
53 \( 1 + (-36.0 + 36.0i)T - 1.48e5iT^{2} \)
59 \( 1 + (106. + 106. i)T + 2.05e5iT^{2} \)
61 \( 1 + (355. + 857. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 - 158.T + 3.00e5T^{2} \)
71 \( 1 + (-536. - 222. i)T + (2.53e5 + 2.53e5i)T^{2} \)
73 \( 1 + (-186. + 449. i)T + (-2.75e5 - 2.75e5i)T^{2} \)
79 \( 1 + (-1.02e3 + 425. i)T + (3.48e5 - 3.48e5i)T^{2} \)
83 \( 1 + (-1.03e3 + 1.03e3i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + (195. - 473. i)T + (-6.45e5 - 6.45e5i)T^{2} \)
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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

−12.54791668679482442441369123405, −12.11319755347895922517323388641, −10.93788168303943406857767352817, −9.384147806884856907817501491562, −8.663306535570823758203376571460, −7.77931031819678472184006285345, −6.37280215354431856232718180577, −5.05568252820361594017861009470, −3.66015506213464349738784065603, −1.94708982368249383095366849493, 0.71937640971803849520897218072, 2.48327013777258260344406148047, 4.35899709980407091685709922895, 5.69589381794241845904240362942, 6.78200447609114429878267908391, 7.937005338184842280296721127564, 9.434242451142644248645391316593, 10.32782285260299049917756563100, 10.90156904884865161989783171430, 12.02985562927873270454337056644

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