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Properties

Label 2-378-63.38-c1-0-7
Degree $2$
Conductor $378$
Sign $-0.810 + 0.586i$
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  − i·2-s − 4-s + (1.82 − 3.15i)5-s + (−1.58 − 2.12i)7-s + i·8-s + (−3.15 − 1.82i)10-s + (−4.38 + 2.53i)11-s + (2.94 − 1.69i)13-s + (−2.12 + 1.58i)14-s + 16-s + (0.774 − 1.34i)17-s + (−0.707 + 0.408i)19-s + (−1.82 + 3.15i)20-s + (2.53 + 4.38i)22-s + (−1.47 − 0.850i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.814 − 1.41i)5-s + (−0.598 − 0.801i)7-s + 0.353i·8-s + (−0.997 − 0.576i)10-s + (−1.32 + 0.763i)11-s + (0.816 − 0.471i)13-s + (−0.566 + 0.422i)14-s + 0.250·16-s + (0.187 − 0.325i)17-s + (−0.162 + 0.0936i)19-s + (−0.407 + 0.705i)20-s + (0.540 + 0.935i)22-s + (−0.307 − 0.177i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.810 + 0.586i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.810 + 0.586i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370279 - 1.14366i\)
\(L(\frac12)\) \(\approx\) \(0.370279 - 1.14366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (1.58 + 2.12i)T \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.38 - 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.774 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.707 - 0.408i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.47 + 0.850i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.60 - 2.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.16iT - 31T^{2} \)
37 \( 1 + (3.39 + 5.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.01 + 1.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 + 5.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 + (-11.4 - 6.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 - 2.45T + 67T^{2} \)
71 \( 1 + 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 - 2.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (0.768 - 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.01 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.59 + 3.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65460243044327195418921774728, −10.22628536911722524052104006667, −9.310750941228266552941230345513, −8.475817357477661488952768033502, −7.39039867388436494110238487754, −5.83652865262535857513208238799, −5.01287400408189906561866514851, −3.92793634909892532193903789762, −2.34074336930004901991439413900, −0.817895918062800246258627651152, 2.45748681633042385678855212814, 3.45955777637659351706978734860, 5.34384332859426166724509076357, 6.14521933380493594344871772083, 6.70294406891208552527302101120, 7.974507002692232783056744050704, 8.879524551621917908186368508173, 9.992097325228831655755009003013, 10.57278837516673370140778218672, 11.59957548293757003674682189474

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