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Properties

Label 2-378-63.31-c2-0-13
Degree $2$
Conductor $378$
Sign $-0.965 + 0.259i$
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  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s − 9.10i·5-s + (−6.61 + 2.29i)7-s + 2.82·8-s + (11.1 + 6.44i)10-s + 6.74·11-s + (4.57 + 2.64i)13-s + (1.85 − 9.72i)14-s + (−2.00 + 3.46i)16-s + (−13.7 − 7.93i)17-s + (−24.4 + 14.0i)19-s + (−15.7 + 9.10i)20-s + (−4.76 + 8.25i)22-s − 7.76·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.82i·5-s + (−0.944 + 0.328i)7-s + 0.353·8-s + (1.11 + 0.644i)10-s + 0.612·11-s + (0.351 + 0.203i)13-s + (0.132 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.808 − 0.466i)17-s + (−1.28 + 0.742i)19-s + (−0.788 + 0.455i)20-s + (−0.216 + 0.375i)22-s − 0.337·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.965 + 0.259i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.965 + 0.259i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0297415 - 0.225670i\)
\(L(\frac12)\) \(\approx\) \(0.0297415 - 0.225670i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 \)
7 \( 1 + (6.61 - 2.29i)T \)
good5 \( 1 + 9.10iT - 25T^{2} \)
11 \( 1 - 6.74T + 121T^{2} \)
13 \( 1 + (-4.57 - 2.64i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (13.7 + 7.93i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (24.4 - 14.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + 7.76T + 529T^{2} \)
29 \( 1 + (-15.3 - 26.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (18.8 - 10.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (9.62 + 16.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (53.2 + 30.7i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-20.2 - 35.1i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (30.6 + 17.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.9 + 27.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (59.8 - 34.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (80.5 + 46.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-48.5 - 84.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 22.5T + 5.04e3T^{2} \)
73 \( 1 + (-50.2 - 28.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-54.8 + 94.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (5.61 - 3.24i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-65.2 + 37.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (32.8 - 18.9i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−10.48462246280616659127467197685, −9.363767372104332404234307132545, −8.904202592783646452646683675629, −8.243210059825153016614704944008, −6.81439414332241191063979454301, −5.95214333237622358608365463999, −4.91100614631350873416254766020, −3.89021397412449170084718529488, −1.67594735881088554739691246962, −0.10992669032084453914385056471, 2.22131130117545432858665487678, 3.26569089456202350564155182024, 4.14375106742815651017580032764, 6.36638589445688663952684803521, 6.61752769727297738822232420947, 7.82045298741540274469943860924, 9.040964699986801773046345177186, 9.999462996168231448872105328311, 10.68836431458428327798470061618, 11.21757669999823743326330048783

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