Properties

Label 2-336-28.3-c1-0-5
Degree $2$
Conductor $336$
Sign $-0.895 + 0.444i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−3 + 1.73i)5-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s + 5.19i·13-s − 3.46i·15-s + (−6 − 3.46i)17-s + (−3.5 − 6.06i)19-s + (2 + 1.73i)21-s + (3.5 − 6.06i)25-s + 0.999·27-s + (−2.5 + 4.33i)31-s + (3 − 1.73i)33-s + (3 + 8.66i)35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−1.34 + 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s + 1.44i·13-s − 0.894i·15-s + (−1.45 − 0.840i)17-s + (−0.802 − 1.39i)19-s + (0.436 + 0.377i)21-s + (0.700 − 1.21i)25-s + 0.192·27-s + (−0.449 + 0.777i)31-s + (0.522 − 0.301i)33-s + (0.507 + 1.46i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (6 + 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.5 - 7.79i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-6 + 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.17658038107463008880336453664, −10.64921731435778230999586349950, −9.304044307619063852762340233950, −8.288780824004539839113294285056, −7.13540886844588892096243509000, −6.65854050248894508109130516911, −4.72203094873673997377963322849, −4.16317481024524719337693887434, −2.83855154370978815589437294842, 0, 2.18035028887262902051783966225, 3.88913670573558239821495281904, 5.07860903041432734059681352532, 5.97477288614643466619648720596, 7.46315815379619051630993934752, 8.214432093735022090255064633657, 8.717449121805435780937217096209, 10.33315490593684826385948079765, 11.14326503702686611059126841332

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