Properties

Label 2-378-21.17-c1-0-9
Degree $2$
Conductor $378$
Sign $-0.507 + 0.861i$
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 + (−1.22 − 2.12i)5-s + (−1 − 2.44i)7-s − 0.999i·8-s + (−2.12 − 1.22i)10-s + (−3.67 − 2.12i)11-s + 4.18i·13-s + (−2.09 − 1.62i)14-s + (−0.5 − 0.866i)16-s + (−1.22 + 2.12i)17-s + (4.24 − 2.44i)19-s − 2.44·20-s − 4.24·22-s + (5.19 − 3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.377 − 0.925i)7-s − 0.353i·8-s + (−0.670 − 0.387i)10-s + (−1.10 − 0.639i)11-s + 1.15i·13-s + (−0.558 − 0.433i)14-s + (−0.125 − 0.216i)16-s + (−0.297 + 0.514i)17-s + (0.973 − 0.561i)19-s − 0.547·20-s − 0.904·22-s + (1.08 − 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.507 + 0.861i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.507 + 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.725775 - 1.27061i\)
\(L(\frac12)\) \(\approx\) \(0.725775 - 1.27061i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.67 + 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.18iT - 13T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 + 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 + 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-3.97 - 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.15 - 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.621 - 0.358i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.74 + 3.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (13.2 + 7.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.62 + 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 + (-8.87 - 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.63iT - 97T^{2} \)
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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

−11.20081687807881202731732554808, −10.33614874097371869673844274233, −9.287233216963825749759262131074, −8.266292611141300835988971129589, −7.24677746307212328023735942723, −6.12783474940424461333674694372, −4.79968519774219734071919844863, −4.18672145210840021439404454344, −2.81855434303915579719796488249, −0.824009368639995085470008281421, 2.70371834603069729774990166611, 3.33064832592204535138309223376, 5.04028860507216640092781304137, 5.73122122292843778804414798441, 7.10180871557025749508183412722, 7.57190389867283249525724145430, 8.757699378618956612764061741524, 9.997297079003790759395636516219, 10.86954330708787085577898820478, 11.77091299779775380806505818143

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