↑

Properties

Label 2-378-189.101-c1-0-21
Degree $2$
Conductor $378$
Sign $-0.500 + 0.865i$
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  + (0.642 − 0.766i)2-s + (0.520 − 1.65i)3-s + (−0.173 − 0.984i)4-s + (1.37 − 1.14i)5-s + (−0.930 − 1.46i)6-s + (−1.38 + 2.25i)7-s + (−0.866 − 0.500i)8-s + (−2.45 − 1.71i)9-s − 1.78i·10-s + (3.15 − 3.75i)11-s + (−1.71 − 0.225i)12-s + (0.0313 − 0.0861i)13-s + (0.839 + 2.50i)14-s + (−1.18 − 2.86i)15-s + (−0.939 + 0.342i)16-s − 0.123·17-s + ⋯
L(s)  = 1  + (0.454 − 0.541i)2-s + (0.300 − 0.953i)3-s + (−0.0868 − 0.492i)4-s + (0.612 − 0.514i)5-s + (−0.380 − 0.596i)6-s + (−0.522 + 0.852i)7-s + (−0.306 − 0.176i)8-s + (−0.819 − 0.573i)9-s − 0.565i·10-s + (0.950 − 1.13i)11-s + (−0.495 − 0.0651i)12-s + (0.00870 − 0.0239i)13-s + (0.224 + 0.670i)14-s + (−0.306 − 0.739i)15-s + (−0.234 + 0.0855i)16-s − 0.0299·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.500 + 0.865i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950261 - 1.64751i\)
\(L(\frac12)\) \(\approx\) \(0.950261 - 1.64751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.642 + 0.766i)T \)
3 \( 1 + (-0.520 + 1.65i)T \)
7 \( 1 + (1.38 - 2.25i)T \)
good5 \( 1 + (-1.37 + 1.14i)T + (0.868 - 4.92i)T^{2} \)
11 \( 1 + (-3.15 + 3.75i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.0313 + 0.0861i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + 0.123T + 17T^{2} \)
19 \( 1 + 2.33iT - 19T^{2} \)
23 \( 1 + (1.95 - 5.37i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.25 - 3.44i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.32 + 0.762i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.72 - 4.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.84 - 2.12i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.72 + 9.80i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.03 + 5.85i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-11.1 - 6.46i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.47 - 2.71i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-8.83 - 1.55i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.05 - 0.886i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (13.4 - 7.77i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.74 - 5.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.96 + 6.68i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3.01 + 1.09i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + (-9.86 - 1.73i)T + (91.1 + 33.1i)T^{2} \)
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

−11.63815391777207628511521586413, −10.09846539087360771723034259111, −8.938274869516578770578979951953, −8.745931536046296091394970609412, −7.08105692007103984050073486037, −6.02046385909697520992912337842, −5.47690945612989655085594510668, −3.64920397508740785948882822507, −2.54558017818342054509300110725, −1.20903577091173013714481571946, 2.51535910592441478296774578784, 3.91256261342220484645543991133, 4.53036930237378043160711579727, 6.00242424417723266443845615126, 6.74761460552738443090722387675, 7.84080913555388434170795248063, 9.050303066065321714314690191744, 9.968801710059204914988172913262, 10.39400330640541807813098121182, 11.66731127423870749056479248960

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