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Properties

Label 2-378-63.32-c2-0-14
Degree $2$
Conductor $378$
Sign $-0.944 - 0.327i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s − 6.30i·5-s + (−3.75 − 5.90i)7-s − 2.82i·8-s + (−4.45 + 7.72i)10-s − 4.87i·11-s + (−1.62 + 2.81i)13-s + (0.428 + 9.89i)14-s + (−2.00 + 3.46i)16-s + (−17.6 − 10.2i)17-s + (14.7 + 25.5i)19-s + (10.9 − 6.30i)20-s + (−3.44 + 5.96i)22-s + 6.32i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 1.26i·5-s + (−0.537 − 0.843i)7-s − 0.353i·8-s + (−0.445 + 0.772i)10-s − 0.443i·11-s + (−0.124 + 0.216i)13-s + (0.0306 + 0.706i)14-s + (−0.125 + 0.216i)16-s + (−1.04 − 0.601i)17-s + (0.775 + 1.34i)19-s + (0.546 − 0.315i)20-s + (−0.156 + 0.271i)22-s + 0.274i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.944 - 0.327i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0831928 + 0.494188i\)
\(L(\frac12)\) \(\approx\) \(0.0831928 + 0.494188i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (3.75 + 5.90i)T \)
good5 \( 1 + 6.30iT - 25T^{2} \)
11 \( 1 + 4.87iT - 121T^{2} \)
13 \( 1 + (1.62 - 2.81i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (17.6 + 10.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 - 25.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 - 6.32iT - 529T^{2} \)
29 \( 1 + (23.6 - 13.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (23.4 + 40.6i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.7 - 44.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (29.7 + 17.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (34.9 + 60.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (10.5 + 6.07i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-2.78 - 1.60i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (91.3 - 52.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.8 - 58.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (13.3 + 23.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 59.5iT - 5.04e3T^{2} \)
73 \( 1 + (-17.7 + 30.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-36.1 + 62.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-110. + 63.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-35.7 + 20.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (73.4 + 127. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.53219986846432211615800394864, −9.589383775553881374546673843453, −8.987120588843445080377694607139, −7.978843914772842783261832583183, −7.08449811360845307769711921424, −5.77564877941650723933659054608, −4.49725546355504534399938773654, −3.41798811266692552436552159582, −1.59149607101021848953649497403, −0.26600523585823481804272277622, 2.19839465275001181304945775367, 3.23379964522398763904184520793, 5.01444655836938501537224101821, 6.31134660584416041856633879993, 6.84056401518978415744362762360, 7.84411518176506847711292401765, 9.062794945486665775774406819858, 9.658066036508055330397274460632, 10.78582172232625108674646176030, 11.27450778179538117865136677163

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