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Properties

Label 2-336-48.35-c1-0-23
Degree $2$
Conductor $336$
Sign $0.100 - 0.994i$
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  + (0.277 + 1.38i)2-s + (1.66 − 0.490i)3-s + (−1.84 + 0.770i)4-s + (2.27 + 2.27i)5-s + (1.14 + 2.16i)6-s + 7-s + (−1.58 − 2.34i)8-s + (2.51 − 1.62i)9-s + (−2.51 + 3.77i)10-s + (−1.28 + 1.28i)11-s + (−2.68 + 2.18i)12-s + (−2.15 − 2.15i)13-s + (0.277 + 1.38i)14-s + (4.88 + 2.65i)15-s + (2.81 − 2.84i)16-s − 6.50i·17-s + ⋯
L(s)  = 1  + (0.196 + 0.980i)2-s + (0.959 − 0.283i)3-s + (−0.922 + 0.385i)4-s + (1.01 + 1.01i)5-s + (0.465 + 0.884i)6-s + 0.377·7-s + (−0.558 − 0.829i)8-s + (0.839 − 0.543i)9-s + (−0.796 + 1.19i)10-s + (−0.388 + 0.388i)11-s + (−0.776 + 0.630i)12-s + (−0.597 − 0.597i)13-s + (0.0742 + 0.370i)14-s + (1.26 + 0.686i)15-s + (0.703 − 0.710i)16-s − 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.100 - 0.994i$
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: \((2,\ 336,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51754 + 1.37223i\)
\(L(\frac12)\) \(\approx\) \(1.51754 + 1.37223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.277 - 1.38i)T \)
3 \( 1 + (-1.66 + 0.490i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.27 - 2.27i)T + 5iT^{2} \)
11 \( 1 + (1.28 - 1.28i)T - 11iT^{2} \)
13 \( 1 + (2.15 + 2.15i)T + 13iT^{2} \)
17 \( 1 + 6.50iT - 17T^{2} \)
19 \( 1 + (3.42 - 3.42i)T - 19iT^{2} \)
23 \( 1 - 5.60iT - 23T^{2} \)
29 \( 1 + (3.59 - 3.59i)T - 29iT^{2} \)
31 \( 1 - 0.730iT - 31T^{2} \)
37 \( 1 + (-7.94 + 7.94i)T - 37iT^{2} \)
41 \( 1 + 3.23T + 41T^{2} \)
43 \( 1 + (8.55 + 8.55i)T + 43iT^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 + (0.785 + 0.785i)T + 53iT^{2} \)
59 \( 1 + (4.42 - 4.42i)T - 59iT^{2} \)
61 \( 1 + (1.23 + 1.23i)T + 61iT^{2} \)
67 \( 1 + (2.62 - 2.62i)T - 67iT^{2} \)
71 \( 1 + 1.31iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 + (-0.603 - 0.603i)T + 83iT^{2} \)
89 \( 1 + 0.657T + 89T^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−12.06867484147716666833697511212, −10.45785318337716496234371227614, −9.703015234309136007714761151916, −8.926890205734773868039383192074, −7.56042300990814647638911408854, −7.28411808319208448037640208500, −6.06537316364073327720199981929, −5.00168301038838732236988657366, −3.44253873087929935140787742102, −2.26714000324792650779546563178, 1.62629952688879127577130382203, 2.58534132745336847248150602844, 4.23115579102829945218356896647, 4.89818613932458024078349505233, 6.19387773813645356128770440547, 8.177680482170674509724077362490, 8.688043174559982139156490677681, 9.581244231547671531564704363023, 10.24261881394803610536905018782, 11.21989913334707451217818514256

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