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Properties

Label 2-378-63.5-c1-0-1
Degree $2$
Conductor $378$
Sign $0.975 - 0.221i$
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 + (1.80 + 3.13i)5-s + (−2.41 − 1.08i)7-s + i·8-s + (3.13 − 1.80i)10-s + (1.73 + 1.00i)11-s + (2.95 + 1.70i)13-s + (−1.08 + 2.41i)14-s + 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 + (2.62 − 1.51i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.809 + 1.40i)5-s + (−0.912 − 0.410i)7-s + 0.353i·8-s + (0.991 − 0.572i)10-s + (0.523 + 0.302i)11-s + (0.818 + 0.472i)13-s + (−0.289 + 0.644i)14-s + 0.250·16-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 + (0.546 − 0.315i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37418 + 0.153942i\)
\(L(\frac12)\) \(\approx\) \(1.37418 + 0.153942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 1.08i)T \)
good5 \( 1 + (-1.80 - 3.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2} \)
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 \( 1 + (-2.62 + 1.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.04 - 2.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.909iT - 31T^{2} \)
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 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + (7.58 - 4.37i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.98T + 59T^{2} \)
61 \( 1 + 14.7iT - 61T^{2} \)
67 \( 1 + 8.31T + 67T^{2} \)
71 \( 1 + 0.466iT - 71T^{2} \)
73 \( 1 + (3.65 - 2.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.82T + 79T^{2} \)
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

−11.00029827151425400564511773526, −10.64640031451619109132425995032, −9.743421755110835065931232578708, −9.056687645786485026090260188713, −7.50795587897510519568278458627, −6.50662956449065236003592595239, −5.82928129302241389557441340873, −3.92985678842807834448127412443, −3.17105754359319213380103691060, −1.77414452530289753096516967526, 1.05793373071623962205814866692, 3.18670920168067340053198294352, 4.71499230281724569828793526691, 5.66580217691117573106653150983, 6.27311433142514931224989456625, 7.63405468790967867423034800477, 8.748800704830143354473554809889, 9.327139796088539739451008364138, 9.930139262853005482758369568570, 11.51450206889008818779781112708

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