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Properties

Label 2-336-48.35-c1-0-29
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: $\chi_{336} (323, \cdot )$
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

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

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