Properties

Label 2-378-21.5-c1-0-5
Degree $2$
Conductor $378$
Sign $0.444 - 0.895i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (2.59 − 1.5i)11-s + 3.46i·13-s + (−0.866 + 2.5i)14-s + (−0.5 + 0.866i)16-s + 3·22-s + (2.5 + 4.33i)25-s + (−1.73 + 2.99i)26-s + (−2 + 1.73i)28-s − 9i·29-s + (−1.5 + 0.866i)31-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (0.783 − 0.452i)11-s + 0.960i·13-s + (−0.231 + 0.668i)14-s + (−0.125 + 0.216i)16-s + 0.639·22-s + (0.5 + 0.866i)25-s + (−0.339 + 0.588i)26-s + (−0.377 + 0.327i)28-s − 1.67i·29-s + (−0.269 + 0.155i)31-s + (−0.153 + 0.0883i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65781 + 1.02851i\)
\(L(\frac12)\) \(\approx\) \(1.65781 + 1.02851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-5.19 + 9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.19T + 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−11.74449814273356112216541761383, −10.91941659870708072117996019779, −9.402286863124858792684289248635, −8.818869199250622480191328647168, −7.69440705983956753760806381873, −6.55560880760996560227572417322, −5.79194262377359026604983492510, −4.67343337505694314375490376297, −3.51379620733472021601159874160, −2.05571940661367282407657899900, 1.28177715324940123000578475017, 3.04912566107652497715812973662, 4.17075565184571896968743572361, 5.10207548236249855531975780740, 6.41730007512408445821643461432, 7.26157105986856312318605980726, 8.380660385625076834093720581221, 9.644486783000991287542290315002, 10.47070200587384476826912916609, 11.15246057101703408429604392646

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