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Properties

Label 2-336-21.20-c1-0-7
Degree $2$
Conductor $336$
Sign $0.692 + 0.721i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
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  + (−0.848 − 1.51i)3-s + 3.02·5-s + (2.56 − 0.662i)7-s + (−1.56 + 2.56i)9-s + 3.12i·11-s − 1.69i·13-s + (−2.56 − 4.56i)15-s + 1.32·17-s − 6.41i·19-s + (−3.17 − 3.30i)21-s + 2i·23-s + 4.12·25-s + (5.19 + 0.185i)27-s − 9.12i·29-s + 7.36i·31-s + ⋯
L(s)  = 1  + (−0.489 − 0.871i)3-s + 1.35·5-s + (0.968 − 0.250i)7-s + (−0.520 + 0.853i)9-s + 0.941i·11-s − 0.470i·13-s + (−0.661 − 1.17i)15-s + 0.321·17-s − 1.47i·19-s + (−0.692 − 0.721i)21-s + 0.417i·23-s + 0.824·25-s + (0.999 + 0.0357i)27-s − 1.69i·29-s + 1.32i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.692 + 0.721i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38045 - 0.588670i\)
\(L(\frac12)\) \(\approx\) \(1.38045 - 0.588670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.848 + 1.51i)T \)
7 \( 1 + (-2.56 + 0.662i)T \)
good5 \( 1 - 3.02T + 5T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
13 \( 1 + 1.69iT - 13T^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 + 6.41iT - 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 9.12iT - 29T^{2} \)
31 \( 1 - 7.36iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 9.12iT - 53T^{2} \)
59 \( 1 + 5.08T + 59T^{2} \)
61 \( 1 - 5.08iT - 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 4.87iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 2.87T + 79T^{2} \)
83 \( 1 + 1.69T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 - 8.68iT - 97T^{2} \)
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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

−11.48278508086759202092655159656, −10.55256874216530556554647835607, −9.726395414536839087093937288605, −8.521897867499839730672729809995, −7.46595664651878335630366004396, −6.62658547368585928797161381312, −5.51239773330024305265824219761, −4.78794072107174549931638044059, −2.48010243869065890304008324598, −1.42461815282329129469324100786, 1.73904844512396491850889186250, 3.46968088782672426703142006458, 4.93099474240491624038286019965, 5.66232270025934445526497911962, 6.44921265733384850256138521913, 8.171786649862266265164850429952, 9.039476277571581568584885394370, 9.928470995904388566770794607241, 10.66774040123442282370411600943, 11.48628597473018194784437332937

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