Properties

Label 2-378-189.41-c1-0-20
Degree $2$
Conductor $378$
Sign $-0.430 + 0.902i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 − 0.173i)2-s + (−0.0750 − 1.73i)3-s + (0.939 − 0.342i)4-s + (−2.14 − 1.80i)5-s + (−0.374 − 1.69i)6-s + (2.53 + 0.743i)7-s + (0.866 − 0.5i)8-s + (−2.98 + 0.259i)9-s + (−2.42 − 1.40i)10-s + (−3.45 − 4.11i)11-s + (−0.662 − 1.60i)12-s + (2.86 + 0.505i)13-s + (2.62 + 0.290i)14-s + (−2.95 + 3.85i)15-s + (0.766 − 0.642i)16-s + (−0.473 + 0.820i)17-s + ⋯
L(s)  = 1  + (0.696 − 0.122i)2-s + (−0.0433 − 0.999i)3-s + (0.469 − 0.171i)4-s + (−0.960 − 0.805i)5-s + (−0.152 − 0.690i)6-s + (0.959 + 0.280i)7-s + (0.306 − 0.176i)8-s + (−0.996 + 0.0865i)9-s + (−0.767 − 0.443i)10-s + (−1.04 − 1.24i)11-s + (−0.191 − 0.461i)12-s + (0.794 + 0.140i)13-s + (0.702 + 0.0777i)14-s + (−0.763 + 0.994i)15-s + (0.191 − 0.160i)16-s + (−0.114 + 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.430 + 0.902i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.430 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.911370 - 1.44512i\)
\(L(\frac12)\) \(\approx\) \(0.911370 - 1.44512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (0.0750 + 1.73i)T \)
7 \( 1 + (-2.53 - 0.743i)T \)
good5 \( 1 + (2.14 + 1.80i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (3.45 + 4.11i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.86 - 0.505i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (0.473 - 0.820i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.211 - 0.122i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.103 - 0.283i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-6.17 + 1.08i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (1.27 + 3.49i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.84 - 4.93i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.505 - 2.86i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-7.36 + 6.18i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-12.6 - 4.58i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 1.40iT - 53T^{2} \)
59 \( 1 + (-8.84 - 7.42i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-4.43 + 12.1i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.239 - 1.35i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (3.49 + 2.01i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (11.8 - 6.83i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.23 - 6.99i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.12 - 12.0i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.16 + 12.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.31 - 2.76i)T + (-16.8 + 95.5i)T^{2} \)
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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

−11.36132417750062104082411651146, −10.71409465719982831831053125945, −8.620436806971383901570553847995, −8.309531098658115510685467814722, −7.42192304039852748125638354555, −6.03323680188779315120463841113, −5.26521427642989811767812272025, −4.06667430638706436504374862721, −2.63446341559204077090746643875, −0.989197879227283345418699964977, 2.60229353896548486529653548301, 3.83432707384265383998430901194, 4.60261592924828917633551695964, 5.53716343111307475668837514702, 7.03103587890858911565277364717, 7.77219355904071090925174589946, 8.763171790944774005523236914231, 10.36882963754988510416534243017, 10.69945573187444305279662497888, 11.54620145623036207340725608539

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