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Properties

Label 2-378-21.2-c2-0-19
Degree $2$
Conductor $378$
Sign $-0.744 + 0.667i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−3.67 − 2.12i)5-s + (5.5 − 4.33i)7-s − 2.82i·8-s + (3 + 5.19i)10-s + 8·13-s + (−9.79 + 1.41i)14-s + (−2.00 + 3.46i)16-s + (−3.67 + 2.12i)17-s + (−2.5 + 4.33i)19-s − 8.48i·20-s + (−18.3 − 10.6i)23-s + (−3.5 − 6.06i)25-s + (−9.79 − 5.65i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.734 − 0.424i)5-s + (0.785 − 0.618i)7-s − 0.353i·8-s + (0.300 + 0.519i)10-s + 0.615·13-s + (−0.699 + 0.101i)14-s + (−0.125 + 0.216i)16-s + (−0.216 + 0.124i)17-s + (−0.131 + 0.227i)19-s − 0.424i·20-s + (−0.798 − 0.461i)23-s + (−0.140 − 0.242i)25-s + (−0.376 − 0.217i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.744 + 0.667i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.283646 - 0.741934i\)
\(L(\frac12)\) \(\approx\) \(0.283646 - 0.741934i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-5.5 + 4.33i)T \)
good5 \( 1 + (3.67 + 2.12i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 - 8T + 169T^{2} \)
17 \( 1 + (3.67 - 2.12i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (18.3 + 10.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 29.6iT - 841T^{2} \)
31 \( 1 + (14.5 + 25.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 72.1iT - 1.68e3T^{2} \)
43 \( 1 + 13T + 1.84e3T^{2} \)
47 \( 1 + (18.3 + 10.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (55.1 - 31.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-29.3 + 16.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (20.5 - 35.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (64 + 110. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 135. iT - 5.04e3T^{2} \)
73 \( 1 + (32.5 + 56.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 67.8iT - 6.88e3T^{2} \)
89 \( 1 + (-91.8 - 53.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 103T + 9.40e3T^{2} \)
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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

−10.82825960995302877243315335653, −10.00293386413479747782822199051, −8.781414195478462469987636673663, −8.103448828985504821716533102988, −7.39939227534539916099673345822, −6.06553350798174523574896004839, −4.51907470807781299356565454025, −3.74389164152463158144003473384, −1.93642696411055100381576062904, −0.44089864888054186034102670734, 1.63663149352882446165380500311, 3.25871329620660509304750882002, 4.71315585326696710053400089974, 5.85804113709738487320534981384, 6.95773067601519847631147121977, 7.892341610094609692517537133850, 8.569754170434402539091320946763, 9.508317171009751013600669693658, 10.73363732188429050301369767180, 11.32647903324271299507498283002

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