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Properties

Label 2-369-41.20-c1-0-14
Degree $2$
Conductor $369$
Sign $0.936 - 0.350i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
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  + (2.29 + 0.746i)2-s + (3.09 + 2.25i)4-s + (1.68 − 2.32i)5-s + (−1.86 − 0.952i)7-s + (2.60 + 3.57i)8-s + (5.61 − 4.07i)10-s + (−0.0609 + 0.384i)11-s + (1.56 + 3.08i)13-s + (−3.58 − 3.58i)14-s + (0.932 + 2.87i)16-s + (−5.05 − 0.799i)17-s + (−2.92 + 5.74i)19-s + (10.4 − 3.40i)20-s + (−0.426 + 0.837i)22-s + (−0.410 + 1.26i)23-s + ⋯
L(s)  = 1  + (1.62 + 0.527i)2-s + (1.54 + 1.12i)4-s + (0.755 − 1.03i)5-s + (−0.706 − 0.360i)7-s + (0.919 + 1.26i)8-s + (1.77 − 1.28i)10-s + (−0.0183 + 0.115i)11-s + (0.435 + 0.854i)13-s + (−0.957 − 0.957i)14-s + (0.233 + 0.717i)16-s + (−1.22 − 0.194i)17-s + (−0.671 + 1.31i)19-s + (2.34 − 0.760i)20-s + (−0.0910 + 0.178i)22-s + (−0.0856 + 0.263i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.936 - 0.350i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.936 - 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.20935 + 0.580065i\)
\(L(\frac12)\) \(\approx\) \(3.20935 + 0.580065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (1.91 + 6.11i)T \)
good2 \( 1 + (-2.29 - 0.746i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (-1.68 + 2.32i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.86 + 0.952i)T + (4.11 + 5.66i)T^{2} \)
11 \( 1 + (0.0609 - 0.384i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-1.56 - 3.08i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (5.05 + 0.799i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (2.92 - 5.74i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (0.410 - 1.26i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.589 - 0.0934i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (0.996 - 0.724i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.07 + 1.51i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-9.42 - 3.06i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.97 + 1.00i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.34 + 0.845i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (2.04 - 6.28i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.08 + 1.97i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.874 - 5.51i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-1.09 + 6.91i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + (-4.23 + 4.23i)T - 79iT^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (0.298 + 0.151i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-2.25 - 14.2i)T + (-92.2 + 29.9i)T^{2} \)
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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

−11.95297265356368310761942582581, −10.75544542709278238769185527690, −9.522931522164863978898555527920, −8.654501775775854734363612662802, −7.21981839482163785801553611034, −6.32755984178489893064366099592, −5.61319753955751492987254416515, −4.51317579376537824195905976714, −3.74201512098490157342852148106, −2.05199577144305561507488283211, 2.35380281567351736366302944389, 2.96896741921761110775120523786, 4.21676216632937578117201717701, 5.50480031451860183229237232884, 6.32053971038928739105492793597, 6.88247288133854554546464592204, 8.708612773445927526295464522762, 9.939453846949018673604623634367, 10.85640772709339737744019426293, 11.25358881779560538877076235024

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