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Properties

Label 2-378-63.59-c1-0-7
Degree $2$
Conductor $378$
Sign $0.962 - 0.271i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 3.61·5-s + (0.266 − 2.63i)7-s + 0.999i·8-s + (3.13 + 1.80i)10-s − 2.00i·11-s + (−2.95 − 1.70i)13-s + (1.54 − 2.14i)14-s + (−0.5 + 0.866i)16-s + (−3.08 + 5.34i)17-s + (0.877 − 0.506i)19-s + (1.80 + 3.13i)20-s + (1.00 − 1.73i)22-s + 3.02i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 1.61·5-s + (0.100 − 0.994i)7-s + 0.353i·8-s + (0.991 + 0.572i)10-s − 0.604i·11-s + (−0.818 − 0.472i)13-s + (0.413 − 0.573i)14-s + (−0.125 + 0.216i)16-s + (−0.748 + 1.29i)17-s + (0.201 − 0.116i)19-s + (0.404 + 0.700i)20-s + (0.213 − 0.369i)22-s + 0.631i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28575 + 0.315661i\)
\(L(\frac12)\) \(\approx\) \(2.28575 + 0.315661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-0.266 + 2.63i)T \)
good5 \( 1 - 3.61T + 5T^{2} \)
11 \( 1 + 2.00iT - 11T^{2} \)
13 \( 1 + (2.95 + 1.70i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.877 + 0.506i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.02iT - 23T^{2} \)
29 \( 1 + (5.04 - 2.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.787 + 0.454i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.66 - 6.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.85 - 4.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.39 + 4.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.58 - 4.37i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.49 + 7.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.7 + 7.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 7.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.466iT - 71T^{2} \)
73 \( 1 + (3.65 + 2.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.00 + 6.93i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.39 + 4.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.1 + 5.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.31372543018174291824842969163, −10.44432051565694960701029563477, −9.734974195658721501171149209343, −8.589426492265606181504524933820, −7.43596782488778591823482650442, −6.43228638717204314304384372856, −5.66681777164212404436441095216, −4.64603392303623749409773641338, −3.24524228551532503145879782290, −1.76935936356884997870380765027, 2.02291483633435285344209940775, 2.61741404869550090988541298991, 4.60913536263014927077599055917, 5.41274282285394590222814353693, 6.26756536347126488549319079723, 7.28874132740565631671659272722, 9.063348085479003961583847857861, 9.472126439123457882296172890276, 10.34969867136269386614687668017, 11.49548411171484798038238109787

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