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Properties

Label 2-378-63.16-c1-0-1
Degree $2$
Conductor $378$
Sign $0.384 - 0.923i$
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 − 1.58·5-s + (−2.64 + 0.0963i)7-s + 0.999·8-s + (0.794 + 1.37i)10-s + 1.58·11-s + (2.40 + 4.16i)13-s + (1.40 + 2.24i)14-s + (−0.5 − 0.866i)16-s + (2.69 + 4.67i)17-s + (−3.54 + 6.14i)19-s + (0.794 − 1.37i)20-s + (−0.794 − 1.37i)22-s − 0.300·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.710·5-s + (−0.999 + 0.0364i)7-s + 0.353·8-s + (0.251 + 0.434i)10-s + 0.478·11-s + (0.667 + 1.15i)13-s + (0.375 + 0.599i)14-s + (−0.125 − 0.216i)16-s + (0.654 + 1.13i)17-s + (−0.814 + 1.41i)19-s + (0.177 − 0.307i)20-s + (−0.169 − 0.293i)22-s − 0.0626·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.384 - 0.923i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.384 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.519194 + 0.346064i\)
\(L(\frac12)\) \(\approx\) \(0.519194 + 0.346064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0963i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.300T + 23T^{2} \)
29 \( 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.35 + 2.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.833 - 1.44i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.33 - 2.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.23 + 5.60i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 8.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.19 + 7.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.18 - 2.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.712 + 1.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54317444974217256483310657108, −10.61904579019137042977631485427, −9.769661439535378098621529712381, −8.847915687339621451727062839948, −8.040453910574620024323675229422, −6.84660254686393453745340411892, −5.89381929066827786350808604127, −3.99317401508695538031094859079, −3.60050624706311344299279063644, −1.72543276160858166446234379667, 0.47917294819184509109412475082, 2.99197892063502348043752662188, 4.19174952634560981867959741388, 5.55940084735348107065927736712, 6.55167411460536584485989717777, 7.42667138031756938272507302555, 8.338357570662908724774989279574, 9.297056691059508565107549911203, 10.08938640986445483370144474859, 11.14733736667289451866259235613

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