Properties

Label 2-336-7.3-c2-0-10
Degree $2$
Conductor $336$
Sign $-0.603 + 0.797i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s + (−1.24 + 0.717i)5-s + (−1.74 + 6.77i)7-s + (1.5 + 2.59i)9-s + (3 − 5.19i)11-s − 21.3i·13-s + 2.48·15-s + (−7.75 − 4.47i)17-s + (6.25 − 3.61i)19-s + (8.48 − 8.66i)21-s + (−18.7 − 32.4i)23-s + (−11.4 + 19.8i)25-s − 5.19i·27-s − 33.9·29-s + (−38.2 − 22.0i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.248 + 0.143i)5-s + (−0.248 + 0.968i)7-s + (0.166 + 0.288i)9-s + (0.272 − 0.472i)11-s − 1.64i·13-s + 0.165·15-s + (−0.456 − 0.263i)17-s + (0.329 − 0.190i)19-s + (0.404 − 0.412i)21-s + (−0.814 − 1.41i)23-s + (−0.458 + 0.794i)25-s − 0.192i·27-s − 1.17·29-s + (−1.23 − 0.711i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.603 + 0.797i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ -0.603 + 0.797i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.279188 - 0.561843i\)
\(L(\frac12)\) \(\approx\) \(0.279188 - 0.561843i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (1.74 - 6.77i)T \)
good5 \( 1 + (1.24 - 0.717i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 21.3iT - 169T^{2} \)
17 \( 1 + (7.75 + 4.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (18.7 + 32.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (38.2 + 22.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 54.8iT - 1.68e3T^{2} \)
43 \( 1 - 1.48T + 1.84e3T^{2} \)
47 \( 1 + (-37.2 + 21.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-42.7 + 74.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-35.6 - 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.02 - 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.19 + 3.80i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 137.T + 5.04e3T^{2} \)
73 \( 1 + (-68.3 - 39.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-49.1 - 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 10.9iT - 9.40e3T^{2} \)
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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

−11.14052378801537224038014951444, −10.21027850519015080806706054815, −9.085401670083490569088607922984, −8.151835481077152472907957382424, −7.14497368805639321203491959667, −5.92285178242963670527150351303, −5.35241684172126701457131833665, −3.68878024531006319202674696792, −2.35070845570825123393360283764, −0.30694248599785445817418134248, 1.65233983664443142631883754336, 3.82836960282743377088542992836, 4.39094462496383554117165315713, 5.83396232702134959754649287498, 6.90785231855669327180410414564, 7.65419972440289389774311525835, 9.159062715593132092511158498251, 9.751919295857267957583741724113, 10.85326984065199690400081492160, 11.57492397230277680847894361117

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