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Properties

Label 2-336-21.20-c3-0-38
Degree $2$
Conductor $336$
Sign $0.585 + 0.810i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.04 − 1.22i)3-s + 10.0·5-s + (−7 − 17.1i)7-s + (23.9 − 12.3i)9-s + 32.9i·11-s − 56.3i·13-s + (50.9 − 12.3i)15-s + 60.5·17-s − 36.7i·19-s + (−56.3 − 78.0i)21-s − 90.7i·23-s − 23.0·25-s + (106. − 91.8i)27-s − 57.7i·29-s + 254. i·31-s + ⋯
L(s)  = 1  + (0.971 − 0.235i)3-s + 0.903·5-s + (−0.377 − 0.925i)7-s + (0.888 − 0.458i)9-s + 0.904i·11-s − 1.20i·13-s + (0.877 − 0.212i)15-s + 0.864·17-s − 0.443i·19-s + (−0.585 − 0.810i)21-s − 0.822i·23-s − 0.184·25-s + (0.755 − 0.654i)27-s − 0.369i·29-s + 1.47i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.585 + 0.810i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.025025436\)
\(L(\frac12)\) \(\approx\) \(3.025025436\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.04 + 1.22i)T \)
7 \( 1 + (7 + 17.1i)T \)
good5 \( 1 - 10.0T + 125T^{2} \)
11 \( 1 - 32.9iT - 1.33e3T^{2} \)
13 \( 1 + 56.3iT - 2.19e3T^{2} \)
17 \( 1 - 60.5T + 4.91e3T^{2} \)
19 \( 1 + 36.7iT - 6.85e3T^{2} \)
23 \( 1 + 90.7iT - 1.21e4T^{2} \)
29 \( 1 + 57.7iT - 2.43e4T^{2} \)
31 \( 1 - 254. iT - 2.97e4T^{2} \)
37 \( 1 - 230T + 5.06e4T^{2} \)
41 \( 1 + 141.T + 6.89e4T^{2} \)
43 \( 1 + 44T + 7.95e4T^{2} \)
47 \( 1 - 343.T + 1.03e5T^{2} \)
53 \( 1 + 206. iT - 1.48e5T^{2} \)
59 \( 1 + 131.T + 2.05e5T^{2} \)
61 \( 1 - 71.0iT - 2.26e5T^{2} \)
67 \( 1 - 64T + 3.00e5T^{2} \)
71 \( 1 + 461. iT - 3.57e5T^{2} \)
73 \( 1 - 88.1iT - 3.89e5T^{2} \)
79 \( 1 - 442T + 4.93e5T^{2} \)
83 \( 1 - 494.T + 5.71e5T^{2} \)
89 \( 1 + 484.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−10.50427664305116736678393002356, −10.06089729230638187803271288067, −9.259254733032624156178663049094, −8.059500235215367915318148444303, −7.26813733460333743412379211801, −6.29642495720275577688200240859, −4.89194015523157895399070417497, −3.56694421176575542308729915485, −2.44024844783761221293071581489, −1.04059053292852101965641029058, 1.68879882454431814503785132631, 2.76337564804176558280672465991, 3.92733849569412393973416897350, 5.46800711806210711603020252554, 6.27522932937359028978668413758, 7.63456037433007556503868990440, 8.671182876164154519218525519547, 9.439105434384422023711564379237, 9.918091991601110354547368910600, 11.24706128170297006816265372002

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