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Properties

Label 2-378-63.59-c1-0-1
Degree $2$
Conductor $378$
Sign $0.215 - 0.976i$
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 − 1.42·5-s + (−2.43 − 1.02i)7-s − 0.999i·8-s + (1.23 + 0.714i)10-s + 3.41i·11-s + (5.48 + 3.16i)13-s + (1.59 + 2.10i)14-s + (−0.5 + 0.866i)16-s + (−1.14 + 1.97i)17-s + (−1.87 + 1.08i)19-s + (−0.714 − 1.23i)20-s + (1.70 − 2.96i)22-s + 8.05i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.639·5-s + (−0.921 − 0.388i)7-s − 0.353i·8-s + (0.391 + 0.226i)10-s + 1.03i·11-s + (1.52 + 0.878i)13-s + (0.426 + 0.563i)14-s + (−0.125 + 0.216i)16-s + (−0.276 + 0.479i)17-s + (−0.430 + 0.248i)19-s + (−0.159 − 0.276i)20-s + (0.364 − 0.631i)22-s + 1.67i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.215 - 0.976i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.215 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470428 + 0.377817i\)
\(L(\frac12)\) \(\approx\) \(0.470428 + 0.377817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.43 + 1.02i)T \)
good5 \( 1 + 1.42T + 5T^{2} \)
11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-5.48 - 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.14 - 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.87 - 1.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.05iT - 23T^{2} \)
29 \( 1 + (-0.298 + 0.172i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.76 - 2.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.07 - 1.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.202 + 0.350i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.75 + 4.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.56 - 4.94i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.51 + 9.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.94 + 5.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.12 + 3.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 + (0.201 + 0.116i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.28 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.811 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.02 + 3.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.42581280623284425382688947656, −10.66891498989482979221030793833, −9.671482824559826063755322731823, −8.974968318303860868354173535876, −7.86340803150294941172374071200, −7.00802946991282967285130639469, −6.06000946509553278854533919652, −4.19276836418590212875594537701, −3.49286560272206456862416744770, −1.67266336912713663724408831249, 0.50164823398011182395198158856, 2.81438540668438447282009505162, 3.99439990810900767028202418926, 5.73827942680721720970833457692, 6.33306224555536220055479641108, 7.52187230189849864042886753952, 8.599797866761186389449808095868, 8.956692424999300481553276403186, 10.36085229335039076204448679016, 10.95190707123722170074169601700

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