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Properties

Label 2-336-48.35-c1-0-5
Degree $2$
Conductor $336$
Sign $0.634 - 0.773i$
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  + (0.230 − 1.39i)2-s + (−0.714 + 1.57i)3-s + (−1.89 − 0.642i)4-s + (1.31 + 1.31i)5-s + (2.03 + 1.36i)6-s − 7-s + (−1.33 + 2.49i)8-s + (−1.97 − 2.25i)9-s + (2.14 − 1.53i)10-s + (−1.23 + 1.23i)11-s + (2.36 − 2.52i)12-s + (4.57 + 4.57i)13-s + (−0.230 + 1.39i)14-s + (−3.02 + 1.13i)15-s + (3.17 + 2.43i)16-s + 5.82i·17-s + ⋯
L(s)  = 1  + (0.162 − 0.986i)2-s + (−0.412 + 0.910i)3-s + (−0.946 − 0.321i)4-s + (0.590 + 0.590i)5-s + (0.831 + 0.555i)6-s − 0.377·7-s + (−0.471 + 0.881i)8-s + (−0.659 − 0.751i)9-s + (0.678 − 0.486i)10-s + (−0.372 + 0.372i)11-s + (0.683 − 0.729i)12-s + (1.26 + 1.26i)13-s + (−0.0615 + 0.372i)14-s + (−0.781 + 0.294i)15-s + (0.793 + 0.608i)16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.634 - 0.773i$
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: \((2,\ 336,\ (\ :1/2),\ 0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.938978 + 0.444262i\)
\(L(\frac12)\) \(\approx\) \(0.938978 + 0.444262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.230 + 1.39i)T \)
3 \( 1 + (0.714 - 1.57i)T \)
7 \( 1 + T \)
good5 \( 1 + (-1.31 - 1.31i)T + 5iT^{2} \)
11 \( 1 + (1.23 - 1.23i)T - 11iT^{2} \)
13 \( 1 + (-4.57 - 4.57i)T + 13iT^{2} \)
17 \( 1 - 5.82iT - 17T^{2} \)
19 \( 1 + (2.27 - 2.27i)T - 19iT^{2} \)
23 \( 1 - 2.63iT - 23T^{2} \)
29 \( 1 + (4.30 - 4.30i)T - 29iT^{2} \)
31 \( 1 + 5.81iT - 31T^{2} \)
37 \( 1 + (-6.99 + 6.99i)T - 37iT^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + (-2.12 - 2.12i)T + 43iT^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + (0.573 + 0.573i)T + 53iT^{2} \)
59 \( 1 + (-8.42 + 8.42i)T - 59iT^{2} \)
61 \( 1 + (6.80 + 6.80i)T + 61iT^{2} \)
67 \( 1 + (-1.90 + 1.90i)T - 67iT^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + 0.516iT - 73T^{2} \)
79 \( 1 - 3.10iT - 79T^{2} \)
83 \( 1 + (-2.91 - 2.91i)T + 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 - 8.09T + 97T^{2} \)
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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

−11.22726094282495756834983758714, −10.94169165096709063065110271801, −9.941357399061942974263583902856, −9.381309526600710897557345778556, −8.335344386738283355631999612994, −6.36506557469525909429511178835, −5.76415546447087541398354246788, −4.28152204149044277173974290587, −3.55714173376699924937730266802, −1.98402416435565952302834726204, 0.75863073102156638943634833464, 3.02044361722202141688537729662, 4.88487083119719600508914628064, 5.73777475071986096469443913782, 6.41756529323122976109238868506, 7.54373251001160243652334374938, 8.400175626389391346267489499436, 9.221938523021942203694712399338, 10.46504915685256396581388204755, 11.61467011603344932839312213192

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