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Properties

Label 2-378-63.58-c1-0-7
Degree $2$
Conductor $378$
Sign $-0.888 - 0.458i$
Analytic cond. $3.01834$
Root an. cond. $1.73733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−1.5 − 2.59i)5-s + (−2 − 1.73i)7-s − 8-s + (1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (−2.5 + 4.33i)13-s + (2 + 1.73i)14-s + 16-s + (1.5 + 2.59i)17-s + (−2.5 + 4.33i)19-s + (−1.5 − 2.59i)20-s + (1.5 − 2.59i)22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (0.474 + 0.821i)10-s + (−0.452 + 0.783i)11-s + (−0.693 + 1.20i)13-s + (0.534 + 0.462i)14-s + 0.250·16-s + (0.363 + 0.630i)17-s + (−0.573 + 0.993i)19-s + (−0.335 − 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (-1.5 - 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.60375925062863853851680724064, −9.836127976125770262399389548979, −9.045628396700969285927040373530, −8.047058264937251934934417284852, −7.31790371965350425379938717046, −6.21773708928578797316902363844, −4.71425038901120794203073840691, −3.80959599410311598621617583258, −1.85724875136122169554801817291, 0, 2.74102370887630031957619182797, 3.28084061081759045993439417565, 5.31324951697940326793430517168, 6.41352881130203080659637104353, 7.32697019660382802746618633381, 8.093802168587464372059090740728, 9.193245987751058934679503617583, 10.12767765604517983346958768164, 10.88709375465246007266692109328

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