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Properties

Label 2-336-16.5-c1-0-14
Degree $2$
Conductor $336$
Sign $-0.729 + 0.683i$
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  + (−1.40 + 0.127i)2-s + (−0.707 + 0.707i)3-s + (1.96 − 0.358i)4-s + (−0.805 − 0.805i)5-s + (0.905 − 1.08i)6-s − i·7-s + (−2.72 + 0.755i)8-s − 1.00i·9-s + (1.23 + 1.03i)10-s + (−0.326 − 0.326i)11-s + (−1.13 + 1.64i)12-s + (−3.46 + 3.46i)13-s + (0.127 + 1.40i)14-s + 1.13·15-s + (3.74 − 1.41i)16-s − 7.57·17-s + ⋯
L(s)  = 1  + (−0.995 + 0.0899i)2-s + (−0.408 + 0.408i)3-s + (0.983 − 0.179i)4-s + (−0.360 − 0.360i)5-s + (0.369 − 0.443i)6-s − 0.377i·7-s + (−0.963 + 0.266i)8-s − 0.333i·9-s + (0.391 + 0.326i)10-s + (−0.0985 − 0.0985i)11-s + (−0.328 + 0.474i)12-s + (−0.960 + 0.960i)13-s + (0.0340 + 0.376i)14-s + 0.294·15-s + (0.935 − 0.352i)16-s − 1.83·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.729 + 0.683i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.729 + 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0724266 - 0.183178i\)
\(L(\frac12)\) \(\approx\) \(0.0724266 - 0.183178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.127i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (0.805 + 0.805i)T + 5iT^{2} \)
11 \( 1 + (0.326 + 0.326i)T + 11iT^{2} \)
13 \( 1 + (3.46 - 3.46i)T - 13iT^{2} \)
17 \( 1 + 7.57T + 17T^{2} \)
19 \( 1 + (-3.34 + 3.34i)T - 19iT^{2} \)
23 \( 1 + 6.11iT - 23T^{2} \)
29 \( 1 + (4.51 - 4.51i)T - 29iT^{2} \)
31 \( 1 + 6.80T + 31T^{2} \)
37 \( 1 + (0.895 + 0.895i)T + 37iT^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (5.85 + 5.85i)T + 43iT^{2} \)
47 \( 1 - 1.34T + 47T^{2} \)
53 \( 1 + (-2.55 - 2.55i)T + 53iT^{2} \)
59 \( 1 + (-5.44 - 5.44i)T + 59iT^{2} \)
61 \( 1 + (6.94 - 6.94i)T - 61iT^{2} \)
67 \( 1 + (-2.62 + 2.62i)T - 67iT^{2} \)
71 \( 1 - 2.61iT - 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 9.66T + 79T^{2} \)
83 \( 1 + (0.419 - 0.419i)T - 83iT^{2} \)
89 \( 1 - 2.60iT - 89T^{2} \)
97 \( 1 + 3.48T + 97T^{2} \)
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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

−11.03186404487497082612739355335, −10.32834056314380497971694468038, −9.149306364114256779032876215539, −8.742787349120255277724945432878, −7.25099264985880940875563983120, −6.74644972139710672946532414443, −5.24853059837348320808973232587, −4.12128092839287213798682365116, −2.28309388621000716025747327476, −0.18487873667878392732307645453, 1.94006298767625094433877408005, 3.30475171169273347687268294968, 5.23978314898066815062856544289, 6.34583315320762367206174069487, 7.42296651947993600928842392794, 7.919737617143200810040381318523, 9.229550064772625392217815559970, 9.983613312511321808125284915020, 11.17786505175443315952990209517, 11.50593152738115867072198757257

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