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Properties

Label 2-378-63.38-c3-0-2
Degree $2$
Conductor $378$
Sign $-0.113 - 0.993i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + (9.84 − 17.0i)5-s + (4.63 + 17.9i)7-s + 8i·8-s + (−34.1 − 19.6i)10-s + (−58.4 + 33.7i)11-s + (−32.3 + 18.6i)13-s + (35.8 − 9.27i)14-s + 16·16-s + (−3.36 + 5.82i)17-s + (−84.3 + 48.7i)19-s + (−39.3 + 68.2i)20-s + (67.4 + 116. i)22-s + (5.29 + 3.05i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.880 − 1.52i)5-s + (0.250 + 0.968i)7-s + 0.353i·8-s + (−1.07 − 0.622i)10-s + (−1.60 + 0.924i)11-s + (−0.689 + 0.397i)13-s + (0.684 − 0.177i)14-s + 0.250·16-s + (−0.0479 + 0.0830i)17-s + (−1.01 + 0.588i)19-s + (−0.440 + 0.762i)20-s + (0.653 + 1.13i)22-s + (0.0479 + 0.0276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.113 - 0.993i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ -0.113 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2969753465\)
\(L(\frac12)\) \(\approx\) \(0.2969753465\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 \)
7 \( 1 + (-4.63 - 17.9i)T \)
good5 \( 1 + (-9.84 + 17.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (58.4 - 33.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (32.3 - 18.6i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (3.36 - 5.82i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (84.3 - 48.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-5.29 - 3.05i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (174. + 101. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 88.0iT - 2.97e4T^{2} \)
37 \( 1 + (-35.4 - 61.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-80.4 - 139. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (256. - 444. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 51.2T + 1.03e5T^{2} \)
53 \( 1 + (261. + 150. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 - 648.T + 2.05e5T^{2} \)
61 \( 1 + 309. iT - 2.26e5T^{2} \)
67 \( 1 + 418.T + 3.00e5T^{2} \)
71 \( 1 + 141. iT - 3.57e5T^{2} \)
73 \( 1 + (-269. - 155. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 266.T + 4.93e5T^{2} \)
83 \( 1 + (383. - 664. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (224. + 388. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-431. - 249. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.28789239956937291740198699987, −9.939959705102245164370617682218, −9.615835403841636326090495304275, −8.560598004454990475970363269538, −7.86053265430825624105832924257, −5.98644058870475536221119813884, −5.11204438838204819541465890286, −4.50947580485896961167163383692, −2.41942608897438647031805564924, −1.77246147301540148125903004030, 0.091609556250674247810222888348, 2.35849321604530804350002096420, 3.46229500002472367932700207947, 5.07555398191233668922337364322, 5.94537168939330868677726050935, 7.03760647150511197062259299694, 7.52110796761800130833790480512, 8.702031509506764123149320524540, 10.07674103942636756111754066651, 10.53682610847851899201794761530

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