Properties

Label 2-378-27.22-c1-0-14
Degree $2$
Conductor $378$
Sign $-0.920 + 0.391i$
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  + (0.766 − 0.642i)2-s + (−1.62 + 0.589i)3-s + (0.173 − 0.984i)4-s + (−1.36 + 0.497i)5-s + (−0.868 + 1.49i)6-s + (−0.173 − 0.984i)7-s + (−0.500 − 0.866i)8-s + (2.30 − 1.92i)9-s + (−0.726 + 1.25i)10-s + (−1.57 − 0.573i)11-s + (0.298 + 1.70i)12-s + (−4.94 − 4.15i)13-s + (−0.766 − 0.642i)14-s + (1.93 − 1.61i)15-s + (−0.939 − 0.342i)16-s + (−2.56 + 4.44i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.940 + 0.340i)3-s + (0.0868 − 0.492i)4-s + (−0.610 + 0.222i)5-s + (−0.354 + 0.611i)6-s + (−0.0656 − 0.372i)7-s + (−0.176 − 0.306i)8-s + (0.768 − 0.640i)9-s + (−0.229 + 0.398i)10-s + (−0.474 − 0.172i)11-s + (0.0860 + 0.492i)12-s + (−1.37 − 1.15i)13-s + (−0.204 − 0.171i)14-s + (0.498 − 0.417i)15-s + (−0.234 − 0.0855i)16-s + (−0.622 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(3.01834\)
Root analytic conductor: \(1.73733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{378} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 378,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106641 - 0.522762i\)
\(L(\frac12)\) \(\approx\) \(0.106641 - 0.522762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (1.62 - 0.589i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
good5 \( 1 + (1.36 - 0.497i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (1.57 + 0.573i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (4.94 + 4.15i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (2.56 - 4.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.75 + 4.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.955 + 5.41i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.505 + 0.423i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.857 + 4.86i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (0.900 - 1.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.89 - 5.78i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.80 - 1.74i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.00405 - 0.0229i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 3.85T + 53T^{2} \)
59 \( 1 + (-5.58 + 2.03i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.16 + 6.61i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (11.9 + 10.0i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (3.86 - 6.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.56 - 9.63i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.30 + 5.29i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (7.97 - 6.69i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (4.13 + 7.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.02 + 1.09i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.89330978704425255010475696899, −10.50320940102683834038158876060, −9.514459106260754659988759737064, −8.021313434540785953137493747566, −6.97487539691731783858597684827, −5.96451793840497946173571724103, −4.84585191076675863043940686665, −4.09984210621433839569283382711, −2.67297371948224888468523439212, −0.31706697879856794769203236539, 2.26501959222970284308088534073, 4.16779275409416115747141944842, 4.96910019416664832349164171264, 5.92189913284083204873437230871, 7.11940992370724040102794092725, 7.55184170600501655403914244366, 8.901111222555936400441808443891, 10.02571878421314092198442569971, 11.19707316897934097705082860954, 12.14514000415197713730325186808

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