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Properties

Label 2-378-63.11-c2-0-2
Degree $2$
Conductor $378$
Sign $-0.209 - 0.977i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
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  + 1.41i·2-s − 2.00·4-s + (−1.22 − 0.707i)5-s + (3.10 − 6.27i)7-s − 2.82i·8-s + (1.00 − 1.73i)10-s + (−15.1 + 8.72i)11-s + (8.59 + 14.8i)13-s + (8.87 + 4.38i)14-s + 4.00·16-s + (20.8 + 12.0i)17-s + (11.7 + 20.4i)19-s + (2.44 + 1.41i)20-s + (−12.3 − 21.3i)22-s + (27.1 + 15.6i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.500·4-s + (−0.244 − 0.141i)5-s + (0.443 − 0.896i)7-s − 0.353i·8-s + (0.100 − 0.173i)10-s + (−1.37 + 0.793i)11-s + (0.660 + 1.14i)13-s + (0.633 + 0.313i)14-s + 0.250·16-s + (1.22 + 0.708i)17-s + (0.620 + 1.07i)19-s + (0.122 + 0.0707i)20-s + (−0.561 − 0.971i)22-s + (1.17 + 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.209 - 0.977i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.209 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.874936 + 1.08170i\)
\(L(\frac12)\) \(\approx\) \(0.874936 + 1.08170i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
7 \( 1 + (-3.10 + 6.27i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (15.1 - 8.72i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.59 - 14.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-20.8 - 12.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.7 - 20.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-27.1 - 15.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.7 - 7.94i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + 32.1T + 961T^{2} \)
37 \( 1 + (-15.7 - 27.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (8.67 - 5.00i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (31.6 - 54.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 4.27iT - 2.20e3T^{2} \)
53 \( 1 + (-45.3 - 26.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + 42.1iT - 3.48e3T^{2} \)
61 \( 1 + 12.4T + 3.72e3T^{2} \)
67 \( 1 - 96.1T + 4.48e3T^{2} \)
71 \( 1 - 20.8iT - 5.04e3T^{2} \)
73 \( 1 + (-41.6 + 72.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 45.1T + 6.24e3T^{2} \)
83 \( 1 + (3.73 + 2.15i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (123. - 71.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (17.5 - 30.3i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.35480055299119204738809685497, −10.33382455502655489071526735220, −9.637135384670201517772231615300, −8.240466045322635164394796125432, −7.74003257049160631153134238168, −6.83335222141154231341932944192, −5.55771735973020408070009711197, −4.60876490535097854513184962473, −3.53823127781911894167682186389, −1.42165843786187790330222829716, 0.69223716979086741616367789181, 2.62588978965216752648830357522, 3.35258494055229096734703245938, 5.23647938876536856424599209641, 5.51150170118763802468012199548, 7.37494007942439185333621207214, 8.256266663238584369292857183627, 9.027092881548092537443169626486, 10.18460460948132166334023647465, 11.02955703331242138086345225115

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