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Properties

Label 2-336-48.35-c1-0-11
Degree $2$
Conductor $336$
Sign $0.613 + 0.789i$
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  + (−1.37 − 0.339i)2-s + (−1.63 − 0.567i)3-s + (1.76 + 0.931i)4-s + (0.132 + 0.132i)5-s + (2.05 + 1.33i)6-s − 7-s + (−2.11 − 1.87i)8-s + (2.35 + 1.85i)9-s + (−0.137 − 0.227i)10-s + (0.715 − 0.715i)11-s + (−2.36 − 2.52i)12-s + (0.206 + 0.206i)13-s + (1.37 + 0.339i)14-s + (−0.142 − 0.292i)15-s + (2.26 + 3.29i)16-s + 6.91i·17-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (−0.944 − 0.327i)3-s + (0.884 + 0.465i)4-s + (0.0594 + 0.0594i)5-s + (0.838 + 0.544i)6-s − 0.377·7-s + (−0.747 − 0.664i)8-s + (0.785 + 0.618i)9-s + (−0.0434 − 0.0719i)10-s + (0.215 − 0.215i)11-s + (−0.683 − 0.729i)12-s + (0.0572 + 0.0572i)13-s + (0.366 + 0.0906i)14-s + (−0.0366 − 0.0755i)15-s + (0.566 + 0.824i)16-s + 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558221 - 0.273056i\)
\(L(\frac12)\) \(\approx\) \(0.558221 - 0.273056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.339i)T \)
3 \( 1 + (1.63 + 0.567i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.132 - 0.132i)T + 5iT^{2} \)
11 \( 1 + (-0.715 + 0.715i)T - 11iT^{2} \)
13 \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \)
17 \( 1 - 6.91iT - 17T^{2} \)
19 \( 1 + (-5.87 + 5.87i)T - 19iT^{2} \)
23 \( 1 + 6.42iT - 23T^{2} \)
29 \( 1 + (-5.00 + 5.00i)T - 29iT^{2} \)
31 \( 1 - 2.52iT - 31T^{2} \)
37 \( 1 + (-6.15 + 6.15i)T - 37iT^{2} \)
41 \( 1 - 5.19T + 41T^{2} \)
43 \( 1 + (-1.36 - 1.36i)T + 43iT^{2} \)
47 \( 1 + 0.603T + 47T^{2} \)
53 \( 1 + (-6.19 - 6.19i)T + 53iT^{2} \)
59 \( 1 + (5.00 - 5.00i)T - 59iT^{2} \)
61 \( 1 + (6.24 + 6.24i)T + 61iT^{2} \)
67 \( 1 + (-2.55 + 2.55i)T - 67iT^{2} \)
71 \( 1 + 0.808iT - 71T^{2} \)
73 \( 1 - 11.5iT - 73T^{2} \)
79 \( 1 - 3.48iT - 79T^{2} \)
83 \( 1 + (-0.641 - 0.641i)T + 83iT^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + 5.56T + 97T^{2} \)
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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

−11.21075328741377408051231256172, −10.57972068556472787554172008840, −9.737616758556285920744178246850, −8.638137775173960056907841108610, −7.62966343318382008911300477230, −6.58396481617446032088496296595, −5.96767268815336414892955236289, −4.28180145211408269573403887071, −2.55508434760470873117440398208, −0.839526588057544375902330587416, 1.17375380158861513406834224227, 3.27934224785195689911289067017, 5.08529989360668063494601604034, 5.89754148320607571793578315216, 7.01229140889442490955167416508, 7.72060050908088497857077177310, 9.394049058800562016485632537678, 9.600410624806950827231948855947, 10.63348214730934671866865168871, 11.69212061931061448605173992276

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