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Properties

Label 2-378-7.2-c3-0-18
Degree $2$
Conductor $378$
Sign $0.481 + 0.876i$
Analytic cond. $22.3027$
Root an. cond. $4.72257$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−5.59 + 9.69i)5-s + (−16.7 + 7.87i)7-s + 7.99·8-s + (−11.1 − 19.3i)10-s + (9.69 + 16.7i)11-s − 39.5·13-s + (3.13 − 36.9i)14-s + (−8 + 13.8i)16-s + (−17.9 − 31.0i)17-s + (−28.0 + 48.6i)19-s + 44.7·20-s − 38.7·22-s + (20.4 − 35.4i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.500 + 0.867i)5-s + (−0.905 + 0.425i)7-s + 0.353·8-s + (−0.354 − 0.613i)10-s + (0.265 + 0.460i)11-s − 0.843·13-s + (0.0597 − 0.704i)14-s + (−0.125 + 0.216i)16-s + (−0.255 − 0.442i)17-s + (−0.339 + 0.587i)19-s + 0.500·20-s − 0.375·22-s + (0.185 − 0.321i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.481 + 0.876i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2743804457\)
\(L(\frac12)\) \(\approx\) \(0.2743804457\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
7 \( 1 + (16.7 - 7.87i)T \)
good5 \( 1 + (5.59 - 9.69i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-9.69 - 16.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 39.5T + 2.19e3T^{2} \)
17 \( 1 + (17.9 + 31.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (28.0 - 48.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-20.4 + 35.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 179.T + 2.43e4T^{2} \)
31 \( 1 + (111. + 193. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (13.6 - 23.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 100.T + 6.89e4T^{2} \)
43 \( 1 + 61.7T + 7.95e4T^{2} \)
47 \( 1 + (-231. + 400. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (174. + 302. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (121. + 211. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (32.5 - 56.4i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (85.4 + 148. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.00e3T + 3.57e5T^{2} \)
73 \( 1 + (573. + 993. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (340. - 590. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 908.T + 5.71e5T^{2} \)
89 \( 1 + (-315. + 546. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−10.55037990594586920069134674681, −9.811554143449391439866531621587, −8.971490289988340097774477701863, −7.80446779962393291634164079818, −6.94880100299320389235340634747, −6.31896640514238056179064576133, −5.00732542164087005404524926707, −3.64790515804719430762761649167, −2.37163233684309076971032413052, −0.12544291981785664860003456181, 1.04732456912772378976196074720, 2.80694422639756788280229813070, 3.97539787840570871065495780072, 4.93950419072810422204216799287, 6.45243890170080740712409635214, 7.49144625594607661057094567127, 8.592922859839964089111355155732, 9.212754668911032710873567579911, 10.20841345992396609789125159341, 11.01686280140588532648757533300

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