Properties

Label 2-378-27.13-c1-0-9
Degree $2$
Conductor $378$
Sign $-0.0794 + 0.996i$
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  + (−0.173 − 0.984i)2-s + (−1.63 − 0.572i)3-s + (−0.939 + 0.342i)4-s + (0.789 + 0.662i)5-s + (−0.279 + 1.70i)6-s + (0.939 + 0.342i)7-s + (0.5 + 0.866i)8-s + (2.34 + 1.87i)9-s + (0.515 − 0.892i)10-s + (0.883 − 0.741i)11-s + (1.73 − 0.0212i)12-s + (1.14 − 6.46i)13-s + (0.173 − 0.984i)14-s + (−0.911 − 1.53i)15-s + (0.766 − 0.642i)16-s + (2.30 − 3.99i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.943 − 0.330i)3-s + (−0.469 + 0.171i)4-s + (0.352 + 0.296i)5-s + (−0.114 + 0.697i)6-s + (0.355 + 0.129i)7-s + (0.176 + 0.306i)8-s + (0.781 + 0.623i)9-s + (0.162 − 0.282i)10-s + (0.266 − 0.223i)11-s + (0.499 − 0.00614i)12-s + (0.316 − 1.79i)13-s + (0.0464 − 0.263i)14-s + (−0.235 − 0.396i)15-s + (0.191 − 0.160i)16-s + (0.559 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0794 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0794 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.0794 + 0.996i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.0794 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.666384 - 0.721574i\)
\(L(\frac12)\) \(\approx\) \(0.666384 - 0.721574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (1.63 + 0.572i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
good5 \( 1 + (-0.789 - 0.662i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (-0.883 + 0.741i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.14 + 6.46i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-2.30 + 3.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.93 - 3.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (8.49 - 3.09i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.0227 + 0.128i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-7.92 + 2.88i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.22 + 3.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.799 + 4.53i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-4.97 + 4.17i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-9.80 - 3.56i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 3.66T + 53T^{2} \)
59 \( 1 + (6.97 + 5.85i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-6.63 - 2.41i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.691 - 3.92i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.62 - 13.2i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.07 - 8.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.154 - 0.878i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.537 + 3.04i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-6.10 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.22 - 4.38i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.15419918328307259630134858793, −10.26588759252815231228838923304, −9.788697951527672246394845881418, −8.192026012922533118495042037274, −7.53369938696470448567090837428, −5.96393091486889884302344035270, −5.48375727473473522936252632071, −4.01599812501268635516833887010, −2.51133283093609623522940595740, −0.896171007083082602006389573331, 1.47873095029066946524144076529, 4.08445940263211436170165293121, 4.77253424882404090066168506625, 6.02232642749333291545608559497, 6.57736827409704024029348332853, 7.76027090730254416016916623226, 8.964614197109336216319864612079, 9.696748443246348777346033042083, 10.60028037205557259405036278931, 11.66811177084034784574792943912

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