Properties

Label 2-153-153.106-c3-0-15
Degree $2$
Conductor $153$
Sign $0.860 - 0.508i$
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  + (−1.93 − 1.11i)2-s + (4.53 + 2.53i)3-s + (−1.49 − 2.58i)4-s + (−12.3 − 3.30i)5-s + (−5.94 − 9.99i)6-s + (−13.8 + 3.72i)7-s + 24.5i·8-s + (14.1 + 23.0i)9-s + (20.2 + 20.2i)10-s + (62.9 − 16.8i)11-s + (−0.203 − 15.5i)12-s + (16.9 + 29.3i)13-s + (31.1 + 8.33i)14-s + (−47.5 − 46.3i)15-s + (15.5 − 26.9i)16-s + (70.0 − 1.33i)17-s + ⋯
L(s)  = 1  + (−0.685 − 0.395i)2-s + (0.872 + 0.488i)3-s + (−0.186 − 0.323i)4-s + (−1.10 − 0.295i)5-s + (−0.404 − 0.680i)6-s + (−0.750 + 0.201i)7-s + 1.08i·8-s + (0.522 + 0.852i)9-s + (0.639 + 0.639i)10-s + (1.72 − 0.462i)11-s + (−0.00489 − 0.373i)12-s + (0.361 + 0.625i)13-s + (0.593 + 0.159i)14-s + (−0.819 − 0.797i)15-s + (0.243 − 0.421i)16-s + (0.999 − 0.0190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.860 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.860 - 0.508i$
Analytic conductor: \(9.02729\)
Root analytic conductor: \(3.00454\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :3/2),\ 0.860 - 0.508i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08059 + 0.295372i\)
\(L(\frac12)\) \(\approx\) \(1.08059 + 0.295372i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.53 - 2.53i)T \)
17 \( 1 + (-70.0 + 1.33i)T \)
good2 \( 1 + (1.93 + 1.11i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (12.3 + 3.30i)T + (108. + 62.5i)T^{2} \)
7 \( 1 + (13.8 - 3.72i)T + (297. - 171.5i)T^{2} \)
11 \( 1 + (-62.9 + 16.8i)T + (1.15e3 - 665.5i)T^{2} \)
13 \( 1 + (-16.9 - 29.3i)T + (-1.09e3 + 1.90e3i)T^{2} \)
19 \( 1 - 124. iT - 6.85e3T^{2} \)
23 \( 1 + (-3.41 + 12.7i)T + (-1.05e4 - 6.08e3i)T^{2} \)
29 \( 1 + (22.0 + 82.1i)T + (-2.11e4 + 1.21e4i)T^{2} \)
31 \( 1 + (-203. - 54.5i)T + (2.57e4 + 1.48e4i)T^{2} \)
37 \( 1 + (195. - 195. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-18.3 + 68.4i)T + (-5.96e4 - 3.44e4i)T^{2} \)
43 \( 1 + (-197. - 113. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (75.4 - 130. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 757. iT - 1.48e5T^{2} \)
59 \( 1 + (-492. + 284. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (325. - 87.3i)T + (1.96e5 - 1.13e5i)T^{2} \)
67 \( 1 + (86.7 + 150. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-269. + 269. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-287. + 287. i)T - 3.89e5iT^{2} \)
79 \( 1 + (461. - 123. i)T + (4.26e5 - 2.46e5i)T^{2} \)
83 \( 1 + (912. + 526. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + (234. + 876. i)T + (-7.90e5 + 4.56e5i)T^{2} \)
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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

−12.27391590862594739644209949346, −11.55406999742823073842255864562, −10.26712017196050676738457692433, −9.434221389195811736438645092957, −8.686205123375764086064946948658, −7.85493781842202156220206471404, −6.15533003894920071296030725528, −4.33424104970839426728476220257, −3.38260732839518234951397816685, −1.32193586722237050481966404990, 0.73939736768578785636704158834, 3.29795058849556040978287591640, 3.98498528729174039460848025977, 6.70197842269966510520776805990, 7.20425442917641288641061001925, 8.233787937255384766489540555433, 9.098145969940253721018836954605, 9.934910200584001076334486397696, 11.63893471417237332920007657063, 12.47434097669760927475886496362

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