Properties

Label 2-378-7.4-c1-0-6
Degree $2$
Conductor $378$
Sign $-0.0633 + 0.997i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.822 − 1.42i)5-s + 2.64·7-s + 0.999·8-s + (−0.822 + 1.42i)10-s + (0.822 − 1.42i)11-s + 0.645·13-s + (−1.32 − 2.29i)14-s + (−0.5 − 0.866i)16-s + (0.822 − 1.42i)17-s + (−1 − 1.73i)19-s + 1.64·20-s − 1.64·22-s + (−4.64 − 8.04i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.368 − 0.637i)5-s + 0.999·7-s + 0.353·8-s + (−0.260 + 0.450i)10-s + (0.248 − 0.429i)11-s + 0.179·13-s + (−0.353 − 0.612i)14-s + (−0.125 − 0.216i)16-s + (0.199 − 0.345i)17-s + (−0.229 − 0.397i)19-s + 0.368·20-s − 0.350·22-s + (−0.968 − 1.67i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.0633 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761368 - 0.811216i\)
\(L(\frac12)\) \(\approx\) \(0.761368 - 0.811216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + (0.822 + 1.42i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.822 + 1.42i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.645T + 13T^{2} \)
17 \( 1 + (-0.822 + 1.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.64 + 8.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 + (0.322 - 0.559i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.96 + 3.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-5.46 - 9.47i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.82 - 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.32 + 10.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.14 - 7.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-5.29 + 9.16i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.61 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 + (-5.46 - 9.47i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.58T + 97T^{2} \)
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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

−11.02200609709205114638738197604, −10.46020033958906419998642294484, −9.131875808979930665446662371854, −8.454783021816057202428604028600, −7.77695139122755657889418922965, −6.35843007795052429720274206397, −4.89003474636887829065215335840, −4.15374233038599652949557590217, −2.53757378193520524351596309838, −0.939764235495284517461314573234, 1.71275111514828229095086366608, 3.61261227028754427283149458135, 4.83614649200238246715473806049, 5.95613676277264001187766085596, 7.06112646755552965656900715277, 7.82857788504580043170789860364, 8.599583532178804732287486178455, 9.786290337793231095702881694129, 10.61592598813957507019349822464, 11.50102337308517826149283513178

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