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Properties

Label 2-378-21.11-c2-0-15
Degree $2$
Conductor $378$
Sign $-0.809 + 0.586i$
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  + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (−0.470 + 0.271i)5-s + (−4.45 − 5.39i)7-s + 2.82i·8-s + (0.384 − 0.666i)10-s + (14.2 + 8.24i)11-s − 8.47·13-s + (9.27 + 3.45i)14-s + (−2.00 − 3.46i)16-s + (−12.3 − 7.14i)17-s + (−11.1 − 19.3i)19-s + 1.08i·20-s − 23.3·22-s + (−24.2 + 13.9i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0941 + 0.0543i)5-s + (−0.637 − 0.770i)7-s + 0.353i·8-s + (0.0384 − 0.0666i)10-s + (1.29 + 0.749i)11-s − 0.652·13-s + (0.662 + 0.246i)14-s + (−0.125 − 0.216i)16-s + (−0.728 − 0.420i)17-s + (−0.587 − 1.01i)19-s + 0.0543i·20-s − 1.06·22-s + (−1.05 + 0.608i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.809 + 0.586i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.809 + 0.586i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0740265 - 0.228222i\)
\(L(\frac12)\) \(\approx\) \(0.0740265 - 0.228222i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (4.45 + 5.39i)T \)
good5 \( 1 + (0.470 - 0.271i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-14.2 - 8.24i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 8.47T + 169T^{2} \)
17 \( 1 + (12.3 + 7.14i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (11.1 + 19.3i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (24.2 - 13.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 10.3iT - 841T^{2} \)
31 \( 1 + (-13.2 + 22.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (26.5 + 45.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 36.7iT - 1.68e3T^{2} \)
43 \( 1 + 45.8T + 1.84e3T^{2} \)
47 \( 1 + (48.9 - 28.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (30.3 + 17.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (53.3 + 30.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (31.7 + 54.9i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.15 + 7.20i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 81.4iT - 5.04e3T^{2} \)
73 \( 1 + (-37.2 + 64.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-40.3 - 69.8i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 129. iT - 6.88e3T^{2} \)
89 \( 1 + (31.6 - 18.2i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 145.T + 9.40e3T^{2} \)
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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

−10.66861728517003718329951265117, −9.512834231741915740626989304603, −9.300655490302756218499618566851, −7.82443366822511237508668227332, −6.97841926051725203331560590863, −6.38677757971015272100606903387, −4.82108511631660446662701166505, −3.69855570451902684114943553715, −1.92978702078645024746887033024, −0.12402297635565386248233941192, 1.81496064534782449448616951605, 3.18614194714434117050239715031, 4.36163257038061256524946303555, 6.10654846505050018281832125741, 6.61296305762544189359416125074, 8.216693905674164318263792326641, 8.684727740193167374706911179429, 9.752547617273011954850763093678, 10.38529542282286576535948063407, 11.78753885869807982564637745943

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