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Properties

Label 2-378-7.2-c3-0-0
Degree $2$
Conductor $378$
Sign $-0.743 - 0.668i$
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 + (−8.97 + 15.5i)5-s + (13.2 − 12.9i)7-s − 7.99·8-s + (17.9 + 31.0i)10-s + (−19.8 − 34.3i)11-s + 28.4·13-s + (−9.22 − 35.8i)14-s + (−8 + 13.8i)16-s + (7.38 + 12.7i)17-s + (−33.9 + 58.7i)19-s + 71.8·20-s − 79.3·22-s + (−88.8 + 153. i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.802 + 1.39i)5-s + (0.714 − 0.699i)7-s − 0.353·8-s + (0.567 + 0.983i)10-s + (−0.543 − 0.941i)11-s + 0.606·13-s + (−0.176 − 0.684i)14-s + (−0.125 + 0.216i)16-s + (0.105 + 0.182i)17-s + (−0.409 + 0.709i)19-s + 0.802·20-s − 0.769·22-s + (−0.805 + 1.39i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.743 - 0.668i$
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: \((2,\ 378,\ (\ :3/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2060909064\)
\(L(\frac12)\) \(\approx\) \(0.2060909064\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.2 + 12.9i)T \)
good5 \( 1 + (8.97 - 15.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (19.8 + 34.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 28.4T + 2.19e3T^{2} \)
17 \( 1 + (-7.38 - 12.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (33.9 - 58.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (88.8 - 153. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 209.T + 2.43e4T^{2} \)
31 \( 1 + (143. + 249. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (183. - 318. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 294.T + 6.89e4T^{2} \)
43 \( 1 + 348.T + 7.95e4T^{2} \)
47 \( 1 + (119. - 206. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (183. + 318. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-127. - 221. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-149. + 258. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-499. - 864. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 659.T + 3.57e5T^{2} \)
73 \( 1 + (319. + 553. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-338. + 586. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 830.T + 5.71e5T^{2} \)
89 \( 1 + (231. - 401. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 373.T + 9.12e5T^{2} \)
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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

−11.33512341346874615712321666776, −10.65732544635163499072494182677, −9.889230674343010214946608532345, −8.260182476995878583878485417209, −7.67678419136027767063057006565, −6.50081312319971138268126958898, −5.40032708900077772523009524258, −3.80220911749361987296092249267, −3.43835058544111494226552412324, −1.80022967933081745604693068960, 0.06000680351829049141800849793, 1.92274628572807976164754840528, 3.83115524516218526794955051737, 4.88009240540968780191196017099, 5.33298926340065127559043439570, 6.84419761495541925600992016544, 7.948437401872367788115699384135, 8.541431291048121058225985720357, 9.239902153782398328992412346727, 10.73470220963338770741407737037

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