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Properties

Label 2-378-189.38-c1-0-17
Degree $2$
Conductor $378$
Sign $0.765 + 0.643i$
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  + (0.984 + 0.173i)2-s + (0.261 − 1.71i)3-s + (0.939 + 0.342i)4-s + (−0.00558 − 0.0316i)5-s + (0.554 − 1.64i)6-s + (2.62 + 0.363i)7-s + (0.866 + 0.5i)8-s + (−2.86 − 0.895i)9-s − 0.0321i·10-s + (2.92 + 0.515i)11-s + (0.831 − 1.51i)12-s + (−0.559 − 0.666i)13-s + (2.51 + 0.812i)14-s + (−0.0556 + 0.00127i)15-s + (0.766 + 0.642i)16-s − 4.59·17-s + ⋯
L(s)  = 1  + (0.696 + 0.122i)2-s + (0.151 − 0.988i)3-s + (0.469 + 0.171i)4-s + (−0.00249 − 0.0141i)5-s + (0.226 − 0.669i)6-s + (0.990 + 0.137i)7-s + (0.306 + 0.176i)8-s + (−0.954 − 0.298i)9-s − 0.0101i·10-s + (0.881 + 0.155i)11-s + (0.240 − 0.438i)12-s + (−0.155 − 0.184i)13-s + (0.672 + 0.217i)14-s + (−0.0143 + 0.000329i)15-s + (0.191 + 0.160i)16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11737 - 0.771311i\)
\(L(\frac12)\) \(\approx\) \(2.11737 - 0.771311i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (-0.261 + 1.71i)T \)
7 \( 1 + (-2.62 - 0.363i)T \)
good5 \( 1 + (0.00558 + 0.0316i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-2.92 - 0.515i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.559 + 0.666i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 4.59T + 17T^{2} \)
19 \( 1 + 4.51iT - 19T^{2} \)
23 \( 1 + (0.272 + 0.324i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.29 - 5.12i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.02 - 2.81i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (2.07 - 3.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.16 + 1.81i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.94 - 1.07i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (6.19 - 2.25i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.47 - 2.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.99 - 3.35i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.208 - 0.572i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-2.60 - 14.7i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (14.2 - 8.20i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-10.4 + 6.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 7.78i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.72 - 2.28i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + 4.71T + 89T^{2} \)
97 \( 1 + (2.81 + 7.72i)T + (-74.3 + 62.3i)T^{2} \)
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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

−11.46750988354673323848058997145, −10.81191036574963392457043533946, −9.063673369406605902902252720568, −8.416961780190388143772655840043, −7.20255316812247793240881114921, −6.66298409313734600959748105114, −5.41006444959332680992979196122, −4.39560309782291296028779416551, −2.82963857248151696834566882150, −1.56662272743605252390447850240, 2.03060654000901252546676164192, 3.63795053506532714427406719848, 4.41446926486894399424843986843, 5.35223429831879216082569220005, 6.46373531138990584694889777217, 7.80054885741925362195302664195, 8.809443878237483537499828436099, 9.711896072185622658149890585139, 10.84417774229459385435445808664, 11.32132872997508655884707895720

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