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Properties

Label 2-387-129.113-c1-0-6
Degree $2$
Conductor $387$
Sign $0.775 - 0.631i$
Analytic cond. $3.09021$
Root an. cond. $1.75789$
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  + (0.0211 + 0.0928i)2-s + (1.79 − 0.863i)4-s + (1.55 + 1.95i)5-s + 1.70i·7-s + (0.237 + 0.297i)8-s + (−0.148 + 0.186i)10-s + (−2.28 + 4.73i)11-s + (0.699 + 0.877i)13-s + (−0.158 + 0.0361i)14-s + (2.46 − 3.08i)16-s + (−0.976 − 0.779i)17-s + (−2.78 − 5.78i)19-s + (4.48 + 2.15i)20-s + (−0.488 − 0.111i)22-s + (1.12 − 2.34i)23-s + ⋯
L(s)  = 1  + (0.0149 + 0.0656i)2-s + (0.896 − 0.431i)4-s + (0.696 + 0.873i)5-s + 0.645i·7-s + (0.0838 + 0.105i)8-s + (−0.0469 + 0.0588i)10-s + (−0.687 + 1.42i)11-s + (0.194 + 0.243i)13-s + (−0.0423 + 0.00967i)14-s + (0.615 − 0.771i)16-s + (−0.236 − 0.188i)17-s + (−0.638 − 1.32i)19-s + (1.00 + 0.482i)20-s + (−0.104 − 0.0237i)22-s + (0.235 − 0.488i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68225 + 0.598433i\)
\(L(\frac12)\) \(\approx\) \(1.68225 + 0.598433i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
43 \( 1 + (-6.55 - 0.132i)T \)
good2 \( 1 + (-0.0211 - 0.0928i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (-1.55 - 1.95i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 - 1.70iT - 7T^{2} \)
11 \( 1 + (2.28 - 4.73i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.699 - 0.877i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (0.976 + 0.779i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 + (2.78 + 5.78i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.12 + 2.34i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.142 - 0.625i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-1.17 - 5.12i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + 5.00iT - 37T^{2} \)
41 \( 1 + (-6.93 + 1.58i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (4.65 + 9.67i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (5.75 + 4.59i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (3.32 + 2.64i)T + (13.1 + 57.5i)T^{2} \)
61 \( 1 + (-8.14 - 1.85i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (12.9 - 6.22i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-6.29 + 3.03i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (7.19 - 5.73i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 7.65T + 79T^{2} \)
83 \( 1 + (2.82 + 0.644i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.41 + 10.5i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-2.17 - 1.04i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.21402531775721287387112455495, −10.54364499597451957500503766945, −9.831141081525418377155927524425, −8.785710839940012241325757587146, −7.27536357763582322484069959121, −6.77912376122996585393547636375, −5.80643005209406005868275220912, −4.73205789317155006526670395332, −2.68744132883877274757402079215, −2.12887625207027601937884538497, 1.34510115003490059317548073374, 2.88116556154753647260472261179, 4.12639776641376609820053032559, 5.67073706903783795269372864038, 6.24082658821772400314790128039, 7.71094227865108369685446035408, 8.266826501136160542708864913042, 9.410488151038461348675680598881, 10.56752296597605195043856417787, 11.05016489795905619651743106288

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