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Properties

Label 2-147-147.104-c1-0-5
Degree $2$
Conductor $147$
Sign $0.994 - 0.105i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (−0.316 − 0.252i)2-s + (0.666 + 1.59i)3-s + (−0.408 − 1.78i)4-s + (1.71 + 0.827i)5-s + (0.192 − 0.674i)6-s + (1.33 − 2.28i)7-s + (−0.674 + 1.40i)8-s + (−2.11 + 2.13i)9-s + (−0.335 − 0.695i)10-s + (3.02 + 2.41i)11-s + (2.58 − 1.84i)12-s + (0.165 + 0.131i)13-s + (−1.00 + 0.386i)14-s + (−0.178 + 3.29i)15-s + (−2.74 + 1.31i)16-s + (0.675 − 2.96i)17-s + ⋯
L(s)  = 1  + (−0.223 − 0.178i)2-s + (0.384 + 0.923i)3-s + (−0.204 − 0.894i)4-s + (0.768 + 0.369i)5-s + (0.0787 − 0.275i)6-s + (0.504 − 0.863i)7-s + (−0.238 + 0.495i)8-s + (−0.704 + 0.710i)9-s + (−0.105 − 0.220i)10-s + (0.912 + 0.727i)11-s + (0.747 − 0.532i)12-s + (0.0457 + 0.0365i)13-s + (−0.267 + 0.103i)14-s + (−0.0460 + 0.851i)15-s + (−0.685 + 0.329i)16-s + (0.163 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $0.994 - 0.105i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 147,\ (\ :1/2),\ 0.994 - 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20473 + 0.0637227i\)
\(L(\frac12)\) \(\approx\) \(1.20473 + 0.0637227i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.666 - 1.59i)T \)
7 \( 1 + (-1.33 + 2.28i)T \)
good2 \( 1 + (0.316 + 0.252i)T + (0.445 + 1.94i)T^{2} \)
5 \( 1 + (-1.71 - 0.827i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (-3.02 - 2.41i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.165 - 0.131i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.675 + 2.96i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 - 0.353iT - 19T^{2} \)
23 \( 1 + (5.70 - 1.30i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.53 + 1.03i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 5.53iT - 31T^{2} \)
37 \( 1 + (1.82 - 7.97i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (6.08 + 2.92i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-5.39 + 2.59i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.89 - 8.64i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (7.60 - 1.73i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (-7.53 + 3.63i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-0.504 - 0.115i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 9.74T + 67T^{2} \)
71 \( 1 + (-12.0 + 2.74i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.11 + 2.48i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 3.31T + 79T^{2} \)
83 \( 1 + (-9.52 - 11.9i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (1.22 + 1.53i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−13.61132860163228223913953621513, −11.66211514056342668084346298749, −10.78028739248444624130203783235, −9.742947777918598096387309880765, −9.586491424544409537488837974000, −8.029993834072288367030142413088, −6.46829294483220282670792931596, −5.17583942613405589212447106297, −4.03339020323570858294362809724, −1.97951559212404913790516827958, 1.91647413733344052160490696933, 3.58593379165023837106055388558, 5.61265611614846663596540676379, 6.65744303956138225777651869578, 8.052227726192984755559746677128, 8.685041788937448665647306593144, 9.474255483281823041022979745431, 11.38896992794239126214560109062, 12.25344268076897073283772319025, 12.95914515352439598190765522123

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