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Properties

Label 2-153-17.13-c3-0-14
Degree $2$
Conductor $153$
Sign $0.999 + 0.0198i$
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  + 5.46i·2-s − 21.8·4-s + (−4.79 − 4.79i)5-s + (3.33 − 3.33i)7-s − 75.9i·8-s + (26.1 − 26.1i)10-s + (6.70 − 6.70i)11-s − 33.7·13-s + (18.2 + 18.2i)14-s + 240.·16-s + (56.0 − 42.0i)17-s − 27.4i·19-s + (104. + 104. i)20-s + (36.6 + 36.6i)22-s + (−58.6 + 58.6i)23-s + ⋯
L(s)  = 1  + 1.93i·2-s − 2.73·4-s + (−0.428 − 0.428i)5-s + (0.180 − 0.180i)7-s − 3.35i·8-s + (0.828 − 0.828i)10-s + (0.183 − 0.183i)11-s − 0.719·13-s + (0.348 + 0.348i)14-s + 3.75·16-s + (0.800 − 0.599i)17-s − 0.331i·19-s + (1.17 + 1.17i)20-s + (0.355 + 0.355i)22-s + (−0.531 + 0.531i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0198i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.999 + 0.0198i$
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: \((2,\ 153,\ (\ :3/2),\ 0.999 + 0.0198i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.646905 - 0.00642265i\)
\(L(\frac12)\) \(\approx\) \(0.646905 - 0.00642265i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-56.0 + 42.0i)T \)
good2 \( 1 - 5.46iT - 8T^{2} \)
5 \( 1 + (4.79 + 4.79i)T + 125iT^{2} \)
7 \( 1 + (-3.33 + 3.33i)T - 343iT^{2} \)
11 \( 1 + (-6.70 + 6.70i)T - 1.33e3iT^{2} \)
13 \( 1 + 33.7T + 2.19e3T^{2} \)
19 \( 1 + 27.4iT - 6.85e3T^{2} \)
23 \( 1 + (58.6 - 58.6i)T - 1.21e4iT^{2} \)
29 \( 1 + (147. + 147. i)T + 2.43e4iT^{2} \)
31 \( 1 + (158. + 158. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-122. - 122. i)T + 5.06e4iT^{2} \)
41 \( 1 + (-60.9 + 60.9i)T - 6.89e4iT^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 + 88.9T + 1.03e5T^{2} \)
53 \( 1 + 541. iT - 1.48e5T^{2} \)
59 \( 1 - 13.5iT - 2.05e5T^{2} \)
61 \( 1 + (112. - 112. i)T - 2.26e5iT^{2} \)
67 \( 1 - 357.T + 3.00e5T^{2} \)
71 \( 1 + (679. + 679. i)T + 3.57e5iT^{2} \)
73 \( 1 + (635. + 635. i)T + 3.89e5iT^{2} \)
79 \( 1 + (319. - 319. i)T - 4.93e5iT^{2} \)
83 \( 1 - 559. iT - 5.71e5T^{2} \)
89 \( 1 - 602.T + 7.04e5T^{2} \)
97 \( 1 + (580. + 580. i)T + 9.12e5iT^{2} \)
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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

−12.91106182867136193334760672763, −11.76367122888496332801554769700, −9.918452051212892296118938540270, −9.084313624452402226022307619422, −7.915645939315351456170013205385, −7.38909632160085430883455013032, −6.05443367879840601375389577786, −5.02879429972937601622010110113, −3.95691150337519481964932372278, −0.32835808503349906816166498420, 1.61199705657706308423653046532, 3.05003763937801976113831078923, 4.11815903194324259153977657394, 5.43995560452246370570328787533, 7.58530470159410558543174248626, 8.795993574878371418187262077987, 9.806944596138288642055253034566, 10.66561312150477007095048194058, 11.48482453084413234717146424376, 12.35912316660546493940516949790

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