Properties

Label 2-336-112.37-c1-0-23
Degree $2$
Conductor $336$
Sign $0.664 - 0.747i$
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  + (1.22 + 0.707i)2-s + (0.965 + 0.258i)3-s + (0.999 + 1.73i)4-s + (1.76 − 0.473i)5-s + (0.999 + i)6-s + (−2.09 − 1.62i)7-s + 2.82i·8-s + (0.866 + 0.499i)9-s + (2.49 + 0.669i)10-s + (−0.107 + 0.400i)11-s + (0.517 + 1.93i)12-s + (0.414 − 0.414i)13-s + (−1.41 − 3.46i)14-s + 1.82·15-s + (−2.00 + 3.46i)16-s + (0.585 + 1.01i)17-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (0.557 + 0.149i)3-s + (0.499 + 0.866i)4-s + (0.789 − 0.211i)5-s + (0.408 + 0.408i)6-s + (−0.790 − 0.612i)7-s + 0.999i·8-s + (0.288 + 0.166i)9-s + (0.789 + 0.211i)10-s + (−0.0323 + 0.120i)11-s + (0.149 + 0.557i)12-s + (0.114 − 0.114i)13-s + (−0.377 − 0.925i)14-s + 0.472·15-s + (−0.500 + 0.866i)16-s + (0.142 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.664 - 0.747i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.664 - 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35793 + 1.05818i\)
\(L(\frac12)\) \(\approx\) \(2.35793 + 1.05818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (2.09 + 1.62i)T \)
good5 \( 1 + (-1.76 + 0.473i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.107 - 0.400i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.414 + 0.414i)T - 13iT^{2} \)
17 \( 1 + (-0.585 - 1.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.732 + 2.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.13 + 4.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.87 + 1.87i)T - 29iT^{2} \)
31 \( 1 + (-1.20 - 2.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.73 - 0.732i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.585iT - 41T^{2} \)
43 \( 1 + (-4.58 - 4.58i)T + 43iT^{2} \)
47 \( 1 + (-3.12 + 5.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.24 + 8.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.536 + 2.00i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.40 - 5.22i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (12.8 + 3.44i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.77 - 8.77i)T - 83iT^{2} \)
89 \( 1 + (-13.4 - 7.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.31T + 97T^{2} \)
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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

−12.02491367491615882475666645611, −10.63804155790730064711328358831, −9.846257307833410877119852554039, −8.768662112375882755253092501383, −7.75732408585719339663177511165, −6.68109317609185109422571576137, −5.88190752910650930420448294176, −4.58556386090157775536473965284, −3.55679932981740707217026072796, −2.27557150376028233453348079510, 1.90375318514325573117241159727, 2.93743263413674387683552713778, 4.06829952622688809698762003048, 5.69500701220429205409893489008, 6.18317267200515584632104823439, 7.43292224903286789246371318857, 8.897655570109539911673668564517, 9.813894410311785633801561684997, 10.34991391570250529088215376877, 11.72961323260012346577479460024

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