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Properties

Label 2-378-7.4-c1-0-2
Degree $2$
Conductor $378$
Sign $-0.605 - 0.795i$
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 + (1.5 + 2.59i)5-s + (−2 + 1.73i)7-s − 0.999·8-s + (−1.5 + 2.59i)10-s − 4·13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s − 3·20-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (−2 − 3.46i)26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 + 1.16i)5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.474 + 0.821i)10-s − 1.10·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s − 0.670·20-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (−0.392 − 0.679i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664470 + 1.34039i\)
\(L(\frac12)\) \(\approx\) \(0.664470 + 1.34039i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.92358990998088304216342319944, −10.65230570450221215349824898259, −9.710471221659176654718370489850, −9.138280791539967093107697946641, −7.49696064333577431235420179954, −7.01591207653807893550005231314, −5.88852546313343139941398961503, −5.20903032460717302478741055474, −3.39487184518830954610970145201, −2.55751672897887108701148856704, 0.932461199028361631564426305531, 2.55527815226750860225589354216, 4.02971640661567738935830614593, 5.00860206016266275502081451712, 5.96820098628800440803179429790, 7.18554859597472449991759226715, 8.519028376233095419155086254444, 9.487501382713376966219188141374, 10.03214955822986258940058106115, 10.97521120396319300884388651646

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