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Properties

Label 2-387-43.37-c2-0-7
Degree $2$
Conductor $387$
Sign $0.907 - 0.420i$
Analytic cond. $10.5449$
Root an. cond. $3.24730$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.26i·2-s − 6.67·4-s + (7.64 + 4.41i)5-s + (−9.63 + 5.56i)7-s + 8.73i·8-s + (14.4 − 24.9i)10-s − 20.1·11-s + (0.195 + 0.338i)13-s + (18.1 + 31.4i)14-s + 1.85·16-s + (9.13 + 15.8i)17-s + (18.7 + 10.8i)19-s + (−51.0 − 29.4i)20-s + 65.7i·22-s + (−12.2 + 21.1i)23-s + ⋯
L(s)  = 1  − 1.63i·2-s − 1.66·4-s + (1.52 + 0.882i)5-s + (−1.37 + 0.794i)7-s + 1.09i·8-s + (1.44 − 2.49i)10-s − 1.82·11-s + (0.0150 + 0.0260i)13-s + (1.29 + 2.24i)14-s + 0.115·16-s + (0.537 + 0.930i)17-s + (0.987 + 0.570i)19-s + (−2.55 − 1.47i)20-s + 2.98i·22-s + (−0.531 + 0.919i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(10.5449\)
Root analytic conductor: \(3.24730\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1),\ 0.907 - 0.420i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.976379 + 0.215204i\)
\(L(\frac12)\) \(\approx\) \(0.976379 + 0.215204i\)
\(L(2)\) 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 \( 1 + (-21.6 + 37.1i)T \)
good2 \( 1 + 3.26iT - 4T^{2} \)
5 \( 1 + (-7.64 - 4.41i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (9.63 - 5.56i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + 20.1T + 121T^{2} \)
13 \( 1 + (-0.195 - 0.338i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-9.13 - 15.8i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-18.7 - 10.8i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (12.2 - 21.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (17.9 - 10.3i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-0.247 + 0.428i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (35.7 + 20.6i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 43.8T + 1.68e3T^{2} \)
47 \( 1 - 34.1T + 2.20e3T^{2} \)
53 \( 1 + (30.6 - 53.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 96.3T + 3.48e3T^{2} \)
61 \( 1 + (24.8 - 14.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.11 + 12.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-5.71 + 3.29i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (30.5 - 17.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (39.8 + 69.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.45 + 5.98i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-113. - 65.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 25.5T + 9.40e3T^{2} \)
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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

−10.78722167400466615696943561560, −10.38055056773233390969590295014, −9.692179543203832993358861146529, −9.119862697671370346452715188833, −7.47065427033293514860292573308, −5.95905183376885191797321992624, −5.50062523751314887569706058436, −3.41388510061146955418586418182, −2.77409275720049958643546867868, −1.86629793355929111365413733359, 0.41145568708426872292568096519, 2.76157939076892641491353292979, 4.72682713072948663944173777836, 5.47876650654973973215207230994, 6.19152541886551389946823055155, 7.18630759430338134007048142387, 8.029090941973681591549701360328, 9.270959997370148069489740670269, 9.705229893014027081630766467246, 10.56449055139317071513635283359

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