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Properties

Label 2-336-336.5-c1-0-23
Degree $2$
Conductor $336$
Sign $-0.290 - 0.957i$
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  + (1.15 + 0.812i)2-s + (0.767 + 1.55i)3-s + (0.680 + 1.88i)4-s + (0.0464 − 0.0124i)5-s + (−0.371 + 2.42i)6-s + (−1.71 − 2.01i)7-s + (−0.739 + 2.73i)8-s + (−1.82 + 2.38i)9-s + (0.0638 + 0.0233i)10-s + (0.714 + 0.191i)11-s + (−2.39 + 2.50i)12-s + (3.92 + 3.92i)13-s + (−0.355 − 3.72i)14-s + (0.0549 + 0.0625i)15-s + (−3.07 + 2.56i)16-s + (−2.15 − 3.73i)17-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)2-s + (0.443 + 0.896i)3-s + (0.340 + 0.940i)4-s + (0.0207 − 0.00556i)5-s + (−0.151 + 0.988i)6-s + (−0.649 − 0.760i)7-s + (−0.261 + 0.965i)8-s + (−0.606 + 0.794i)9-s + (0.0202 + 0.00737i)10-s + (0.215 + 0.0577i)11-s + (−0.691 + 0.722i)12-s + (1.08 + 1.08i)13-s + (−0.0950 − 0.995i)14-s + (0.0141 + 0.0161i)15-s + (−0.768 + 0.640i)16-s + (−0.523 − 0.906i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.290 - 0.957i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.290 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32454 + 1.78553i\)
\(L(\frac12)\) \(\approx\) \(1.32454 + 1.78553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.812i)T \)
3 \( 1 + (-0.767 - 1.55i)T \)
7 \( 1 + (1.71 + 2.01i)T \)
good5 \( 1 + (-0.0464 + 0.0124i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.714 - 0.191i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.92 - 3.92i)T + 13iT^{2} \)
17 \( 1 + (2.15 + 3.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.52 + 1.21i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.18 + 5.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.72 + 5.72i)T + 29iT^{2} \)
31 \( 1 + (-7.88 + 4.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.91 + 1.31i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.05iT - 41T^{2} \)
43 \( 1 + (-0.936 - 0.936i)T + 43iT^{2} \)
47 \( 1 + (-3.32 + 5.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.831 + 0.222i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.850 + 3.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.88 + 1.30i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.17 + 1.92i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 + (-0.701 - 1.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.88 - 8.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.54 - 4.54i)T - 83iT^{2} \)
89 \( 1 + (4.11 + 2.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.21iT - 97T^{2} \)
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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

−11.59848948528394103927219069346, −11.20784188341169293038998767573, −9.784607708024188187892778294456, −9.086903886682440282947482120673, −7.950242565154693945755804128666, −6.90734501084474585557888159453, −5.94312636558716036492410593150, −4.55352230795909038048735925122, −3.92182095865705544031087265450, −2.74376599452241714669585241835, 1.39452498492497852932348262211, 2.88387840670467253812853486355, 3.67627199652136930625692540569, 5.61249410320614900487562396740, 6.10540881743103428241390020516, 7.28562666163614778458098406018, 8.557144860621319305247447747152, 9.419597232393608857397523368233, 10.55944672251482665072173100581, 11.60665507511911339530873143660

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