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Properties

Label 2-336-21.17-c1-0-3
Degree $2$
Conductor $336$
Sign $0.0633 - 0.997i$
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  + (−0.866 + 1.5i)3-s + (0.866 + 1.5i)5-s + (2.5 − 0.866i)7-s + (−1.5 − 2.59i)9-s + (2.59 + 1.5i)11-s + 3.46i·13-s − 3·15-s + (−1.73 + 3i)17-s + (−3 + 1.73i)19-s + (−0.866 + 4.5i)21-s + (−5.19 + 3i)23-s + (1 − 1.73i)25-s + 5.19·27-s − 3i·29-s + (1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.387 + 0.670i)5-s + (0.944 − 0.327i)7-s + (−0.5 − 0.866i)9-s + (0.783 + 0.452i)11-s + 0.960i·13-s − 0.774·15-s + (−0.420 + 0.727i)17-s + (−0.688 + 0.397i)19-s + (−0.188 + 0.981i)21-s + (−1.08 + 0.625i)23-s + (0.200 − 0.346i)25-s + 1.00·27-s − 0.557i·29-s + (0.269 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923479 + 0.866733i\)
\(L(\frac12)\) \(\approx\) \(0.923479 + 0.866733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.46 + 6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.66T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−11.54191354918443289023554965269, −10.85072435730083692796846023604, −10.05925886673206445798570390637, −9.202703077090712347135118964781, −8.086120145241144515194198703059, −6.70890832276067215807618294663, −6.00076783053197475316371868028, −4.56842575147709646841048389374, −3.89850917740199073788705009239, −1.96256300809264731868526966723, 1.04784546751905315534518203223, 2.44087620048846942370061215593, 4.52399597871888444051520260104, 5.50529876586778317533940923165, 6.33635423387891132184298810170, 7.59966122631732781563735896525, 8.439130717757164395285216676361, 9.256417938828949617537840206594, 10.74695345821353941289802100056, 11.36184169510788185212568578533

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