↑

Properties

Label 2-378-63.41-c1-0-3
Degree $2$
Conductor $378$
Sign $0.937 - 0.349i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.183 − 0.317i)5-s + (2.53 + 0.744i)7-s + 0.999i·8-s + 0.366i·10-s + (−0.579 + 0.334i)11-s + (−0.867 − 0.500i)13-s + (−2.57 + 0.624i)14-s + (−0.5 − 0.866i)16-s + 4.98·17-s − 6.35i·19-s + (−0.183 − 0.317i)20-s + (0.334 − 0.579i)22-s + (6.66 + 3.84i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.0819 − 0.141i)5-s + (0.959 + 0.281i)7-s + 0.353i·8-s + 0.115i·10-s + (−0.174 + 0.100i)11-s + (−0.240 − 0.138i)13-s + (−0.687 + 0.166i)14-s + (−0.125 − 0.216i)16-s + 1.21·17-s − 1.45i·19-s + (−0.0409 − 0.0709i)20-s + (0.0713 − 0.123i)22-s + (1.38 + 0.802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.937 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13827 + 0.205254i\)
\(L(\frac12)\) \(\approx\) \(1.13827 + 0.205254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 - 0.744i)T \)
good5 \( 1 + (-0.183 + 0.317i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.579 - 0.334i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.867 + 0.500i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 + 6.35iT - 19T^{2} \)
23 \( 1 + (-6.66 - 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.58 - 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.47 - 3.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + (-2.15 + 3.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.16 + 7.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (4.36 - 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.29 - 2.47i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.49iT - 71T^{2} \)
73 \( 1 - 4.07iT - 73T^{2} \)
79 \( 1 + (4.17 + 7.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.50 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (14.9 - 8.60i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.25897743619956101375510455395, −10.50911627296248919477014664695, −9.370376480127836454361933072046, −8.709361626591006834705837435159, −7.67005319885440774711551264948, −6.96409082717538211209792573705, −5.47934588537768856731180770843, −4.87226590234558240017505397228, −2.95922330103898370400592643816, −1.32212933553603649494884239653, 1.28767326149428325860116780692, 2.79943301808077527856410302836, 4.21693529328365378201631669510, 5.44941335447050474398304602001, 6.76416689107483104352431311529, 7.87085446205058438581787253583, 8.383103481902086286350291332305, 9.649406611472499273718878260291, 10.40866351532651625857378941581, 11.14853516685117892596650956790

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