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Properties

Label 2-336-48.11-c1-0-44
Degree $2$
Conductor $336$
Sign $-0.220 - 0.975i$
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  + (−1 − i)2-s + (−1 − 1.41i)3-s + 2i·4-s + (0.414 − 0.414i)5-s + (−0.414 + 2.41i)6-s − 7-s + (2 − 2i)8-s + (−1.00 + 2.82i)9-s − 0.828·10-s + (−3.82 − 3.82i)11-s + (2.82 − 2i)12-s + (−4.41 + 4.41i)13-s + (1 + i)14-s + (−1 − 0.171i)15-s − 4·16-s − 1.17i·17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.577 − 0.816i)3-s + i·4-s + (0.185 − 0.185i)5-s + (−0.169 + 0.985i)6-s − 0.377·7-s + (0.707 − 0.707i)8-s + (−0.333 + 0.942i)9-s − 0.261·10-s + (−1.15 − 1.15i)11-s + (0.816 − 0.577i)12-s + (−1.22 + 1.22i)13-s + (0.267 + 0.267i)14-s + (−0.258 − 0.0442i)15-s − 16-s − 0.284i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.220 - 0.975i)\)

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 \( 1 + (1 + i)T \)
3 \( 1 + (1 + 1.41i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.414 + 0.414i)T - 5iT^{2} \)
11 \( 1 + (3.82 + 3.82i)T + 11iT^{2} \)
13 \( 1 + (4.41 - 4.41i)T - 13iT^{2} \)
17 \( 1 + 1.17iT - 17T^{2} \)
19 \( 1 + (-0.414 - 0.414i)T + 19iT^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + (-1.82 - 1.82i)T + 37iT^{2} \)
41 \( 1 + 0.343T + 41T^{2} \)
43 \( 1 + (3.82 - 3.82i)T - 43iT^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + (5.82 - 5.82i)T - 53iT^{2} \)
59 \( 1 + (10.0 + 10.0i)T + 59iT^{2} \)
61 \( 1 + (-2.41 + 2.41i)T - 61iT^{2} \)
67 \( 1 + (7 + 7i)T + 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 5.17iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-8.89 + 8.89i)T - 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.25530663010486111005269923466, −9.922919865543001847449229067231, −9.229726167061955302290347299005, −7.907216741793766775223809057454, −7.37146131685154598764561839780, −6.09512333626927729927754599689, −4.89285331386396891369699667375, −3.10795733396149695554613528780, −1.83390511900531902714314170877, 0, 2.65273695317407281741397417057, 4.69124886545526011313388607680, 5.33002939648882355782904676472, 6.48391214539481755884914310108, 7.42806485205728782308563659016, 8.503019450685924762509530264929, 9.679148007132511332450094878662, 10.32812553328061650153823437119, 10.62603969049315088651723658680

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