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Properties

Label 2-378-9.5-c2-0-2
Degree $2$
Conductor $378$
Sign $-0.524 - 0.851i$
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 + (5.52 + 3.18i)5-s + (1.32 + 2.29i)7-s + 2.82i·8-s − 9.01·10-s + (−14.5 + 8.42i)11-s + (−8.77 + 15.1i)13-s + (−3.24 − 1.87i)14-s + (−2.00 − 3.46i)16-s − 13.4i·17-s + 34.1·19-s + (11.0 − 6.37i)20-s + (11.9 − 20.6i)22-s + (3.08 + 1.77i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.10 + 0.637i)5-s + (0.188 + 0.327i)7-s + 0.353i·8-s − 0.901·10-s + (−1.32 + 0.765i)11-s + (−0.674 + 1.16i)13-s + (−0.231 − 0.133i)14-s + (−0.125 − 0.216i)16-s − 0.790i·17-s + 1.79·19-s + (0.552 − 0.318i)20-s + (0.541 − 0.937i)22-s + (0.133 + 0.0773i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(10.2997\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.577622 + 1.03387i\)
\(L(\frac12)\) \(\approx\) \(0.577622 + 1.03387i\)
\(L(2)\) 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 \( 1 + (-5.52 - 3.18i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (14.5 - 8.42i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.77 - 15.1i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 13.4iT - 289T^{2} \)
19 \( 1 - 34.1T + 361T^{2} \)
23 \( 1 + (-3.08 - 1.77i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (29.4 - 16.9i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (15.0 - 25.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 37.7T + 1.36e3T^{2} \)
41 \( 1 + (28.4 + 16.4i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-34.9 - 60.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-22.9 + 13.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 8.14iT - 2.80e3T^{2} \)
59 \( 1 + (-87.5 - 50.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-8.32 - 14.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (21.0 - 36.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 14.4iT - 5.04e3T^{2} \)
73 \( 1 + 42.4T + 5.32e3T^{2} \)
79 \( 1 + (27.1 + 47.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-33.4 + 19.2i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 98.5iT - 7.92e3T^{2} \)
97 \( 1 + (-10.5 - 18.3i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.27284198957566784307458385307, −10.22429445544924799747938267247, −9.680112137645946644725217586917, −8.913977512668594957258170548195, −7.38409367080111958117690474828, −7.07754013511362997846862198891, −5.64106711654469900250975191570, −5.00035645100121317975832922342, −2.82030204117613288232213736736, −1.82554878400739858506275272153, 0.62035914302964803583937169357, 2.10793387859030851424252505545, 3.37744409837616765166855896189, 5.24883518772427996041734077420, 5.70697922630538991853021187146, 7.41251824560333209319021580329, 8.090985884676459829805131873505, 9.139251272875893742541558025208, 10.02628720161195204915131611162, 10.53740407723130756459999360699

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