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Properties

Label 2-378-9.2-c2-0-3
Degree $2$
Conductor $378$
Sign $-0.546 - 0.837i$
Analytic cond. $10.2997$
Root an. cond. $3.20932$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.76 + 1.59i)5-s + (−1.32 + 2.29i)7-s + 2.82i·8-s − 4.52·10-s + (16.3 + 9.42i)11-s + (−5.18 − 8.98i)13-s + (−3.24 + 1.87i)14-s + (−2.00 + 3.46i)16-s + 24.4i·17-s − 26.5·19-s + (−5.53 − 3.19i)20-s + (13.3 + 23.0i)22-s + (−35.3 + 20.4i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.553 + 0.319i)5-s + (−0.188 + 0.327i)7-s + 0.353i·8-s − 0.452·10-s + (1.48 + 0.857i)11-s + (−0.398 − 0.690i)13-s + (−0.231 + 0.133i)14-s + (−0.125 + 0.216i)16-s + 1.43i·17-s − 1.39·19-s + (−0.276 − 0.159i)20-s + (0.606 + 1.04i)22-s + (−1.53 + 0.888i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.859910 + 1.58848i\)
\(L(\frac12)\) \(\approx\) \(0.859910 + 1.58848i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (1.32 - 2.29i)T \)
good5 \( 1 + (2.76 - 1.59i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-16.3 - 9.42i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (5.18 + 8.98i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 24.4iT - 289T^{2} \)
19 \( 1 + 26.5T + 361T^{2} \)
23 \( 1 + (35.3 - 20.4i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-18.8 - 10.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.37 - 2.38i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 45.6T + 1.36e3T^{2} \)
41 \( 1 + (-12.8 + 7.41i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-14.4 + 25.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-25.7 - 14.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 18.1iT - 2.80e3T^{2} \)
59 \( 1 + (-46.4 + 26.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.9 + 24.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-46.2 - 80.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 81.9iT - 5.04e3T^{2} \)
73 \( 1 + 81.2T + 5.32e3T^{2} \)
79 \( 1 + (-42.3 + 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-99.9 - 57.6i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 13.7iT - 7.92e3T^{2} \)
97 \( 1 + (-55.7 + 96.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.73181380762234285971752951156, −10.65436713821963614341371386539, −9.644240513248039281949470446225, −8.488679736738365225259706610669, −7.60993602364513569991661365111, −6.57832034980445841745324668706, −5.82977346687116212704800361538, −4.31303294682314144760989925792, −3.66378221783401273700937270921, −2.01557069065687750494162425522, 0.65541276554297921822338088408, 2.45191645351587662093823153297, 4.02218389379000074961400630471, 4.43137248644363140464020484142, 6.08561568614483987834769648176, 6.75921944057073285962841907237, 8.078853421685210390584783316183, 9.081695401103944056057155919020, 9.983238017693224712275664705607, 11.14091347616742730448873671030

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