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Properties

Label 2-378-21.17-c3-0-18
Degree $2$
Conductor $378$
Sign $0.516 + 0.856i$
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  + (1.73 − i)2-s + (1.99 − 3.46i)4-s + (−3.80 − 6.59i)5-s + (18.4 + 1.98i)7-s − 7.99i·8-s + (−13.1 − 7.61i)10-s + (35.9 + 20.7i)11-s + 69.6i·13-s + (33.8 − 14.9i)14-s + (−8 − 13.8i)16-s + (54.3 − 94.1i)17-s + (82.5 − 47.6i)19-s − 30.4·20-s + 82.9·22-s + (−109. + 63.3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.340 − 0.589i)5-s + (0.994 + 0.106i)7-s − 0.353i·8-s + (−0.416 − 0.240i)10-s + (0.984 + 0.568i)11-s + 1.48i·13-s + (0.646 − 0.286i)14-s + (−0.125 − 0.216i)16-s + (0.775 − 1.34i)17-s + (0.996 − 0.575i)19-s − 0.340·20-s + 0.804·22-s + (−0.994 + 0.574i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(22.3027\)
Root analytic conductor: \(4.72257\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :3/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.075492504\)
\(L(\frac12)\) \(\approx\) \(3.075492504\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
3 \( 1 \)
7 \( 1 + (-18.4 - 1.98i)T \)
good5 \( 1 + (3.80 + 6.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-35.9 - 20.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 69.6iT - 2.19e3T^{2} \)
17 \( 1 + (-54.3 + 94.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-82.5 + 47.6i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (109. - 63.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 208. iT - 2.43e4T^{2} \)
31 \( 1 + (243. + 140. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-113. - 195. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 446.T + 6.89e4T^{2} \)
43 \( 1 - 228.T + 7.95e4T^{2} \)
47 \( 1 + (139. + 241. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (118. + 68.1i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-405. + 702. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (489. - 282. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (475. - 822. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 448. iT - 3.57e5T^{2} \)
73 \( 1 + (-267. - 154. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-139. - 242. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 495.T + 5.71e5T^{2} \)
89 \( 1 + (248. + 429. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 389. iT - 9.12e5T^{2} \)
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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

−11.38148894634567409581489142565, −9.670222083642303903141728782393, −9.244114440377045721271418327181, −7.87159392628496809813044433724, −7.02197481034638583748071222636, −5.68585526861071145312415298148, −4.62138936619242186713305042541, −4.01559184785017087852053964881, −2.25310479062920526641101665376, −1.05090520516394993679611947495, 1.35502204819204762366471760214, 3.18826576942320053035395371791, 3.96524667247014531248876759557, 5.39985808800189402943356723921, 6.08949493387084065318975145478, 7.49916765115025544761719668394, 7.928413749355905582703384434064, 9.065497151306482458634499748818, 10.65920939759974985127473655914, 10.92961541680163174635287791937

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