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Properties

Label 2-378-21.20-c1-0-2
Degree $2$
Conductor $378$
Sign $-0.755 - 0.654i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
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  + i·2-s − 4-s − 1.73·5-s + (2 + 1.73i)7-s − i·8-s − 1.73i·10-s + 3i·11-s + 3.46i·13-s + (−1.73 + 2i)14-s + 16-s − 6.92·17-s + 1.73i·19-s + 1.73·20-s − 3·22-s + 3i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.774·5-s + (0.755 + 0.654i)7-s − 0.353i·8-s − 0.547i·10-s + 0.904i·11-s + 0.960i·13-s + (−0.462 + 0.534i)14-s + 0.250·16-s − 1.68·17-s + 0.397i·19-s + 0.387·20-s − 0.639·22-s + 0.625i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336354 + 0.902178i\)
\(L(\frac12)\) \(\approx\) \(0.336354 + 0.902178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 3iT - 71T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−11.65829347179685834056416686401, −11.03691457841884819142619487537, −9.555270959883553525954986347738, −8.860577689329856263759497452619, −7.88175738467019999506348158549, −7.13760509092754352747438475199, −6.04563459151078904046672920376, −4.75680924392216707947067771664, −4.08282557117265537752476112023, −2.09103980985007047506957425150, 0.65742208539132196488344514468, 2.58436935133502546713499710304, 3.94386026353510527376359751277, 4.70629313770613856950238198729, 6.12390283099408262641506106817, 7.54394187372462970620798312318, 8.248020643847419198410929074308, 9.135122218821324976242156531872, 10.49208491166796264591504321122, 11.05352991001634789015774597396

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