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Properties

Label 2-336-7.5-c2-0-2
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: $\chi_{336} (145, \cdot )$
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

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

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