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Properties

Label 2-336-16.5-c1-0-9
Degree $2$
Conductor $336$
Sign $-0.0230 - 0.999i$
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  + (1.25 + 0.659i)2-s + (−0.707 + 0.707i)3-s + (1.13 + 1.65i)4-s + (2.66 + 2.66i)5-s + (−1.35 + 0.418i)6-s − i·7-s + (0.325 + 2.80i)8-s − 1.00i·9-s + (1.57 + 5.08i)10-s + (−4.19 − 4.19i)11-s + (−1.96 − 0.367i)12-s + (0.146 − 0.146i)13-s + (0.659 − 1.25i)14-s − 3.76·15-s + (−1.44 + 3.72i)16-s − 1.52·17-s + ⋯
L(s)  = 1  + (0.884 + 0.466i)2-s + (−0.408 + 0.408i)3-s + (0.565 + 0.825i)4-s + (1.19 + 1.19i)5-s + (−0.551 + 0.170i)6-s − 0.377i·7-s + (0.115 + 0.993i)8-s − 0.333i·9-s + (0.498 + 1.60i)10-s + (−1.26 − 1.26i)11-s + (−0.567 − 0.106i)12-s + (0.0406 − 0.0406i)13-s + (0.176 − 0.334i)14-s − 0.972·15-s + (−0.361 + 0.932i)16-s − 0.369·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.0230 - 0.999i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.0230 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52144 + 1.55688i\)
\(L(\frac12)\) \(\approx\) \(1.52144 + 1.55688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.659i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-2.66 - 2.66i)T + 5iT^{2} \)
11 \( 1 + (4.19 + 4.19i)T + 11iT^{2} \)
13 \( 1 + (-0.146 + 0.146i)T - 13iT^{2} \)
17 \( 1 + 1.52T + 17T^{2} \)
19 \( 1 + (-5.84 + 5.84i)T - 19iT^{2} \)
23 \( 1 - 5.92iT - 23T^{2} \)
29 \( 1 + (-1.34 + 1.34i)T - 29iT^{2} \)
31 \( 1 + 0.0732T + 31T^{2} \)
37 \( 1 + (4.05 + 4.05i)T + 37iT^{2} \)
41 \( 1 + 1.55iT - 41T^{2} \)
43 \( 1 + (3.52 + 3.52i)T + 43iT^{2} \)
47 \( 1 - 9.26T + 47T^{2} \)
53 \( 1 + (-1.23 - 1.23i)T + 53iT^{2} \)
59 \( 1 + (-0.548 - 0.548i)T + 59iT^{2} \)
61 \( 1 + (-5.91 + 5.91i)T - 61iT^{2} \)
67 \( 1 + (1.40 - 1.40i)T - 67iT^{2} \)
71 \( 1 + 4.40iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + (9.09 - 9.09i)T - 83iT^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 + 0.580T + 97T^{2} \)
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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

−11.56318974441745544817019182372, −10.95211662647837289590883955673, −10.26373951216278896381030945502, −9.057634083716457003287697871438, −7.58250274435636057372231321631, −6.77274034516419921839182144281, −5.71725141797575707173856614746, −5.22494564138071395732575452070, −3.47852478442850303763031455386, −2.60621327650192671875320895883, 1.48356895958890646506455387750, 2.52555033162025722929956107722, 4.59605046194871051012354064137, 5.29843814459632962441091759024, 5.99217534835575088602091416731, 7.25105042145305289691847073164, 8.603663184041944561227575824477, 9.958609003504311701410941303714, 10.19293225166366979373650094393, 11.68525946123836143943565511387

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