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Properties

Label 2-378-189.101-c1-0-18
Degree $2$
Conductor $378$
Sign $-0.331 + 0.943i$
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  + (0.642 − 0.766i)2-s + (−1.53 − 0.792i)3-s + (−0.173 − 0.984i)4-s + (2.60 − 2.18i)5-s + (−1.59 + 0.669i)6-s + (2.61 − 0.388i)7-s + (−0.866 − 0.500i)8-s + (1.74 + 2.44i)9-s − 3.40i·10-s + (1.28 − 1.53i)11-s + (−0.513 + 1.65i)12-s + (−1.44 + 3.97i)13-s + (1.38 − 2.25i)14-s + (−5.74 + 1.30i)15-s + (−0.939 + 0.342i)16-s − 4.69·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (−0.889 − 0.457i)3-s + (−0.0868 − 0.492i)4-s + (1.16 − 0.977i)5-s + (−0.652 + 0.273i)6-s + (0.989 − 0.146i)7-s + (−0.306 − 0.176i)8-s + (0.580 + 0.814i)9-s − 1.07i·10-s + (0.387 − 0.461i)11-s + (−0.148 + 0.477i)12-s + (−0.401 + 1.10i)13-s + (0.369 − 0.602i)14-s + (−1.48 + 0.335i)15-s + (−0.234 + 0.0855i)16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.949529 - 1.34001i\)
\(L(\frac12)\) \(\approx\) \(0.949529 - 1.34001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (1.53 + 0.792i)T \)
7 \( 1 + (-2.61 + 0.388i)T \)
good5 \( 1 + (-2.60 + 2.18i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-1.28 + 1.53i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (1.44 - 3.97i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 4.69T + 17T^{2} \)
19 \( 1 + 3.54iT - 19T^{2} \)
23 \( 1 + (-1.82 + 5.00i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.76 - 4.86i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (1.74 - 0.308i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (4.63 - 8.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.602 + 0.219i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.43 - 8.14i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.290 - 1.64i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-10.2 - 5.89i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.95 - 1.07i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (14.5 + 2.56i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.63 - 3.88i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-11.4 + 6.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.54 + 5.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0278 + 0.0233i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-1.70 + 0.621i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 4.82T + 89T^{2} \)
97 \( 1 + (-4.01 - 0.708i)T + (91.1 + 33.1i)T^{2} \)
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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

−11.20930763890999719237201318588, −10.49776349229382679709321410755, −9.274985396758957421726356077103, −8.582689115453835458387748969187, −6.92723716452694933813355045609, −6.13394592119197232624639885383, −4.84309022657502825550627942515, −4.69011470267427476510325454994, −2.17981121054852443550267145909, −1.21910236611785938915813826746, 2.12797273943796926304243183261, 3.81525438778182482695805813973, 5.14115044843267836444682835115, 5.71909765692775279968835010962, 6.68820519292186892724817573512, 7.55230325474967554471735930957, 8.997470933799447621493917020763, 10.03640848397053229165417631065, 10.70341642609733479761144425557, 11.57331799368591941243811239982

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