Properties

Label 2-378-63.47-c1-0-5
Degree $2$
Conductor $378$
Sign $0.603 + 0.797i$
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 + 0.900·5-s + (1.05 − 2.42i)7-s − 0.999i·8-s + (0.779 − 0.450i)10-s + 3.12i·11-s + (1.99 − 1.14i)13-s + (−0.296 − 2.62i)14-s + (−0.5 − 0.866i)16-s + (−2.57 − 4.46i)17-s + (2.38 + 1.37i)19-s + (0.450 − 0.779i)20-s + (1.56 + 2.70i)22-s − 1.71i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.402·5-s + (0.399 − 0.916i)7-s − 0.353i·8-s + (0.246 − 0.142i)10-s + 0.943i·11-s + (0.552 − 0.318i)13-s + (−0.0793 − 0.702i)14-s + (−0.125 − 0.216i)16-s + (−0.624 − 1.08i)17-s + (0.546 + 0.315i)19-s + (0.100 − 0.174i)20-s + (0.333 + 0.577i)22-s − 0.357i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85642 - 0.922986i\)
\(L(\frac12)\) \(\approx\) \(1.85642 - 0.922986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.05 + 2.42i)T \)
good5 \( 1 - 0.900T + 5T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
13 \( 1 + (-1.99 + 1.14i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.57 + 4.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.71iT - 23T^{2} \)
29 \( 1 + (-1.85 - 1.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.66 - 5.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.73 - 8.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.22 - 2.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.273 - 0.473i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.93 + 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (12.0 - 6.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.99 - 6.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.28 + 3.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.83 - 3.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (10.9 - 6.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.27 - 5.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.184 + 0.319i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.00 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.86 + 5.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34897894317779376923814075040, −10.27097618215778613509743821106, −9.810836593982248227105531382584, −8.427547118985610016764451277055, −7.27164395643307736690315879462, −6.43916285860344866123688989915, −5.09041545348200731554497494687, −4.33492489754238979412203740323, −2.95501656635692146890086538250, −1.42775615912404417010488136947, 2.01272219160916294164943177452, 3.40809654108578699273488808687, 4.70259720677952632127644902848, 5.88959323993234802331553120356, 6.29979504018152296384607518454, 7.84385716623453074788800403114, 8.599012391351584774352102610122, 9.507382695445347123461077572383, 10.88399859671850641908518184351, 11.53592546171841679532823528758

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