Properties

Label 2-336-48.35-c1-0-1
Degree $2$
Conductor $336$
Sign $-0.0430 - 0.999i$
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  + (−1.15 − 0.810i)2-s + (−1.50 + 0.861i)3-s + (0.686 + 1.87i)4-s + (−1.19 − 1.19i)5-s + (2.43 + 0.219i)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 + (−2.64 − 2.23i)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 + (−0.867 + 0.497i)3-s + (0.343 + 0.939i)4-s + (−0.536 − 0.536i)5-s + (0.995 + 0.0897i)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 + (−0.764 − 0.644i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0430 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.0430 - 0.999i$
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: \((2,\ 336,\ (\ :1/2),\ -0.0430 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251985 + 0.263080i\)
\(L(\frac12)\) \(\approx\) \(0.251985 + 0.263080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 - 0.861i)T \)
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

−11.68256424255957977853385347866, −10.69478215016910836222674512539, −10.25905359768570264172013006377, −9.035478476668041277715802620973, −8.265061084550581900554365636298, −7.20969351783786297116999846153, −5.93960783767797650070777619006, −4.55291497166183514109168449948, −3.67782259730462889070989365287, −1.56556970429272903465134209795, 0.37818485410838890889472947642, 2.33978871744282169123749905038, 4.58186840054548880324709904329, 5.68967965849388648272213401900, 6.68105606557913519942399910882, 7.44349638247713276621369074630, 8.221793015475550246519534340454, 9.418138125641123353067625715922, 10.65158224568642713811974543926, 11.12708415781181512888050927320

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