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Properties

Label 2-378-63.5-c1-0-5
Degree $2$
Conductor $378$
Sign $0.322 + 0.946i$
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 + (−0.714 − 1.23i)5-s + (0.327 − 2.62i)7-s − i·8-s + (1.23 − 0.714i)10-s + (−2.96 − 1.70i)11-s + (−5.48 − 3.16i)13-s + (2.62 + 0.327i)14-s + 16-s + (1.14 + 1.97i)17-s + (−1.87 − 1.08i)19-s + (0.714 + 1.23i)20-s + (1.70 − 2.96i)22-s + (6.97 − 4.02i)23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.319 − 0.553i)5-s + (0.123 − 0.992i)7-s − 0.353i·8-s + (0.391 − 0.226i)10-s + (−0.892 − 0.515i)11-s + (−1.52 − 0.878i)13-s + (0.701 + 0.0875i)14-s + 0.250·16-s + (0.276 + 0.479i)17-s + (−0.430 − 0.248i)19-s + (0.159 + 0.276i)20-s + (0.364 − 0.631i)22-s + (1.45 − 0.839i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $0.322 + 0.946i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ 0.322 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681209 - 0.487739i\)
\(L(\frac12)\) \(\approx\) \(0.681209 - 0.487739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.327 + 2.62i)T \)
good5 \( 1 + (0.714 + 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.96 + 1.70i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.48 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.14 - 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.97 + 4.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.298 + 0.172i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.34iT - 31T^{2} \)
37 \( 1 + (-1.07 + 1.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.202 - 0.350i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 + (8.56 - 4.94i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 4.25T + 67T^{2} \)
71 \( 1 + 3.55iT - 71T^{2} \)
73 \( 1 + (0.201 - 0.116i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + (0.811 + 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.02 + 3.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.89042267712811021220260069378, −10.35501960601269195002443111021, −9.190697734968860189725934801825, −8.097719381617847698946140062902, −7.60208063954081866521313621144, −6.52254318308467557483802454128, −5.13956117576900526803532078631, −4.55698413227681996752726985014, −3.00277909207768925217243129419, −0.55023626632798784548096733094, 2.16560931751070197001173168068, 3.04525836264425286376308347360, 4.63187709084111055823992625788, 5.44749440445329305451472935876, 7.00475280353688951589812203584, 7.78792575053529812043952249965, 9.103495716333893814451315425411, 9.692721047468772172947719495814, 10.76713095446298271653342942617, 11.57500361588008342029049009800

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