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Properties

Label 2-378-63.38-c1-0-0
Degree $2$
Conductor $378$
Sign $-0.955 + 0.295i$
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 + (−0.483 + 0.837i)5-s + (−2.16 − 1.52i)7-s − i·8-s + (−0.837 − 0.483i)10-s + (−4.82 + 2.78i)11-s + (−3.76 + 2.17i)13-s + (1.52 − 2.16i)14-s + 16-s + (−1.97 + 3.41i)17-s + (3.86 − 2.23i)19-s + (0.483 − 0.837i)20-s + (−2.78 − 4.82i)22-s + (−2.29 − 1.32i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.216 + 0.374i)5-s + (−0.817 − 0.576i)7-s − 0.353i·8-s + (−0.264 − 0.152i)10-s + (−1.45 + 0.840i)11-s + (−1.04 + 0.603i)13-s + (0.407 − 0.577i)14-s + 0.250·16-s + (−0.478 + 0.828i)17-s + (0.887 − 0.512i)19-s + (0.108 − 0.187i)20-s + (−0.594 − 1.02i)22-s + (−0.479 − 0.276i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.955 + 0.295i$
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: \((2,\ 378,\ (\ :1/2),\ -0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0535370 - 0.354488i\)
\(L(\frac12)\) \(\approx\) \(0.0535370 - 0.354488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 1.52i)T \)
good5 \( 1 + (0.483 - 0.837i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.82 - 2.78i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.76 - 2.17i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.97 - 3.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.86 + 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.29 + 1.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.61 + 2.66i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.16iT - 31T^{2} \)
37 \( 1 + (-0.243 - 0.421i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0818 + 0.141i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.35 - 7.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.49T + 47T^{2} \)
53 \( 1 + (-1.74 - 1.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.67T + 59T^{2} \)
61 \( 1 - 5.17iT - 61T^{2} \)
67 \( 1 + 5.44T + 67T^{2} \)
71 \( 1 - 3.64iT - 71T^{2} \)
73 \( 1 + (2.15 + 1.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 4.60T + 79T^{2} \)
83 \( 1 + (-4.20 + 7.29i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.05 - 3.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.2 - 5.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.95995320928214129532972765336, −10.72795329222198792978081503065, −9.942317217917102253892004027635, −9.209967301995303168398538977633, −7.68187358190750835968375027015, −7.35449412095044469532551800532, −6.30160665797772307351520733745, −5.08241034412128154827750445195, −4.03573718877397011301024767661, −2.56849464845644249258105486712, 0.22348145138094437118950414230, 2.51647809689493517527046063217, 3.36650191483939417873937418086, 5.01933594260112089387006917238, 5.64719961820838662697863845133, 7.24219886237454516987607683579, 8.246346405617854810041670471306, 9.155123325986740874541525421226, 10.05580009300451913061264756169, 10.77467346541836675687622678837

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