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Properties

Label 2-387-43.15-c1-0-5
Degree $2$
Conductor $387$
Sign $0.682 - 0.730i$
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  + (0.651 − 0.816i)2-s + (0.202 + 0.885i)4-s + (0.0373 + 0.498i)5-s + (−1.65 + 2.86i)7-s + (2.73 + 1.31i)8-s + (0.431 + 0.294i)10-s + (−0.828 + 3.62i)11-s + (−3.87 + 2.64i)13-s + (1.26 + 3.21i)14-s + (1.22 − 0.589i)16-s + (0.343 − 4.57i)17-s + (4.44 − 4.12i)19-s + (−0.433 + 0.133i)20-s + (2.42 + 3.04i)22-s + (0.371 − 0.114i)23-s + ⋯
L(s)  = 1  + (0.460 − 0.577i)2-s + (0.101 + 0.442i)4-s + (0.0167 + 0.222i)5-s + (−0.624 + 1.08i)7-s + (0.967 + 0.466i)8-s + (0.136 + 0.0930i)10-s + (−0.249 + 1.09i)11-s + (−1.07 + 0.732i)13-s + (0.337 + 0.859i)14-s + (0.305 − 0.147i)16-s + (0.0831 − 1.11i)17-s + (1.01 − 0.945i)19-s + (−0.0970 + 0.0299i)20-s + (0.517 + 0.648i)22-s + (0.0774 − 0.0239i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(387\)    =    \(3^{2} \cdot 43\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(3.09021\)
Root analytic conductor: \(1.75789\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{387} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 387,\ (\ :1/2),\ 0.682 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49022 + 0.647098i\)
\(L(\frac12)\) \(\approx\) \(1.49022 + 0.647098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.26 + 1.92i)T \)
good2 \( 1 + (-0.651 + 0.816i)T + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (-0.0373 - 0.498i)T + (-4.94 + 0.745i)T^{2} \)
7 \( 1 + (1.65 - 2.86i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.828 - 3.62i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.87 - 2.64i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-0.343 + 4.57i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-4.44 + 4.12i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-0.371 + 0.114i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (-1.29 - 3.29i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-0.861 - 0.129i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.72 + 8.43i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.776 - 3.40i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (3.94 + 2.68i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-6.05 + 2.91i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (5.18 - 0.782i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (7.73 - 7.17i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (0.859 + 0.265i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (0.798 - 0.544i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-0.500 + 0.867i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.24 + 3.17i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-0.672 + 1.71i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (0.213 - 0.936i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.76391778732876886532437631348, −10.71866904596047992910488231013, −9.559187484377172096705893189497, −8.986421854146851644576320295424, −7.40636949972946444926159031519, −6.98420929019411063187902810657, −5.31327387971230206622930874934, −4.51546754832940097915016630696, −2.93859761457674019711517970826, −2.34981860239585280899843650335, 0.985523902875408131351986822112, 3.18288091397667383746232403495, 4.39573825324205600222099778645, 5.53247375293284405918821088297, 6.28854525120057119837338348019, 7.37267811750902788925319897916, 8.104905050417031479111735218895, 9.622537498351731348911655935659, 10.30461723370486847278150310169, 10.96117131281131542135864912389

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