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Properties

Label 2-336-48.35-c1-0-14
Degree $2$
Conductor $336$
Sign $-0.542 - 0.840i$
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  + (1.15 + 0.810i)2-s + (−0.861 + 1.50i)3-s + (0.686 + 1.87i)4-s + (1.19 + 1.19i)5-s + (−2.21 + 1.04i)6-s + 7-s + (−0.726 + 2.73i)8-s + (−1.51 − 2.58i)9-s + (0.417 + 2.36i)10-s + (1.10 − 1.10i)11-s + (−3.41 − 0.585i)12-s + (0.418 + 0.418i)13-s + (1.15 + 0.810i)14-s + (−2.83 + 0.769i)15-s + (−3.05 + 2.57i)16-s − 4.01i·17-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)2-s + (−0.497 + 0.867i)3-s + (0.343 + 0.939i)4-s + (0.536 + 0.536i)5-s + (−0.904 + 0.426i)6-s + 0.377·7-s + (−0.256 + 0.966i)8-s + (−0.505 − 0.862i)9-s + (0.132 + 0.746i)10-s + (0.332 − 0.332i)11-s + (−0.985 − 0.169i)12-s + (0.116 + 0.116i)13-s + (0.309 + 0.216i)14-s + (−0.731 + 0.198i)15-s + (−0.764 + 0.644i)16-s − 0.974i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.542 - 0.840i$
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: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.542 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917649 + 1.68456i\)
\(L(\frac12)\) \(\approx\) \(0.917649 + 1.68456i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.810i)T \)
3 \( 1 + (0.861 - 1.50i)T \)
7 \( 1 - T \)
good5 \( 1 + (-1.19 - 1.19i)T + 5iT^{2} \)
11 \( 1 + (-1.10 + 1.10i)T - 11iT^{2} \)
13 \( 1 + (-0.418 - 0.418i)T + 13iT^{2} \)
17 \( 1 + 4.01iT - 17T^{2} \)
19 \( 1 + (4.91 - 4.91i)T - 19iT^{2} \)
23 \( 1 + 3.26iT - 23T^{2} \)
29 \( 1 + (-4.61 + 4.61i)T - 29iT^{2} \)
31 \( 1 - 7.31iT - 31T^{2} \)
37 \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \)
41 \( 1 - 8.46T + 41T^{2} \)
43 \( 1 + (-3.91 - 3.91i)T + 43iT^{2} \)
47 \( 1 + 8.08T + 47T^{2} \)
53 \( 1 + (-1.78 - 1.78i)T + 53iT^{2} \)
59 \( 1 + (-5.55 + 5.55i)T - 59iT^{2} \)
61 \( 1 + (-0.325 - 0.325i)T + 61iT^{2} \)
67 \( 1 + (1.41 - 1.41i)T - 67iT^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 - 2.89iT - 73T^{2} \)
79 \( 1 + 15.6iT - 79T^{2} \)
83 \( 1 + (-9.34 - 9.34i)T + 83iT^{2} \)
89 \( 1 + 6.70T + 89T^{2} \)
97 \( 1 + 3.43T + 97T^{2} \)
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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

−11.87558351368360007187462863059, −11.01338879081392692003581543181, −10.23279321693109722778721618420, −9.035238166013345410250355479365, −8.033335735676257864764783457924, −6.56534107935890228531377839557, −6.06647375158127408932421170203, −4.90031341107422873241840578057, −3.99585520348730720055707396589, −2.66446802941304209950418796561, 1.27781966937305985350655390375, 2.39813805389776806613832433238, 4.25293574339324726883545896333, 5.29796184581437117692864235115, 6.14301980611987136990405225131, 7.07464012441292740174511464599, 8.429382974752674059533799505618, 9.545550440483334380298204903972, 10.73965752890424333024719135490, 11.34504109565002219529010379850

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