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Properties

Label 2-336-48.11-c1-0-40
Degree $2$
Conductor $336$
Sign $0.531 + 0.847i$
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  + (1.14 − 0.827i)2-s + (1.73 − 0.0404i)3-s + (0.630 − 1.89i)4-s + (0.0573 − 0.0573i)5-s + (1.95 − 1.47i)6-s − 7-s + (−0.847 − 2.69i)8-s + (2.99 − 0.139i)9-s + (0.0183 − 0.113i)10-s + (1.96 + 1.96i)11-s + (1.01 − 3.31i)12-s + (−4.48 + 4.48i)13-s + (−1.14 + 0.827i)14-s + (0.0969 − 0.101i)15-s + (−3.20 − 2.39i)16-s − 4.71i·17-s + ⋯
L(s)  = 1  + (0.810 − 0.585i)2-s + (0.999 − 0.0233i)3-s + (0.315 − 0.949i)4-s + (0.0256 − 0.0256i)5-s + (0.797 − 0.603i)6-s − 0.377·7-s + (−0.299 − 0.954i)8-s + (0.998 − 0.0466i)9-s + (0.00579 − 0.0358i)10-s + (0.592 + 0.592i)11-s + (0.292 − 0.956i)12-s + (−1.24 + 1.24i)13-s + (−0.306 + 0.221i)14-s + (0.0250 − 0.0262i)15-s + (−0.801 − 0.598i)16-s − 1.14i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.531 + 0.847i$
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: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.531 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29556 - 1.27005i\)
\(L(\frac12)\) \(\approx\) \(2.29556 - 1.27005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 + 0.827i)T \)
3 \( 1 + (-1.73 + 0.0404i)T \)
7 \( 1 + T \)
good5 \( 1 + (-0.0573 + 0.0573i)T - 5iT^{2} \)
11 \( 1 + (-1.96 - 1.96i)T + 11iT^{2} \)
13 \( 1 + (4.48 - 4.48i)T - 13iT^{2} \)
17 \( 1 + 4.71iT - 17T^{2} \)
19 \( 1 + (2.60 + 2.60i)T + 19iT^{2} \)
23 \( 1 - 2.30iT - 23T^{2} \)
29 \( 1 + (-3.24 - 3.24i)T + 29iT^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + (2.13 + 2.13i)T + 37iT^{2} \)
41 \( 1 + 6.69T + 41T^{2} \)
43 \( 1 + (-1.86 + 1.86i)T - 43iT^{2} \)
47 \( 1 - 5.14T + 47T^{2} \)
53 \( 1 + (-1.91 + 1.91i)T - 53iT^{2} \)
59 \( 1 + (3.22 + 3.22i)T + 59iT^{2} \)
61 \( 1 + (7.78 - 7.78i)T - 61iT^{2} \)
67 \( 1 + (-8.73 - 8.73i)T + 67iT^{2} \)
71 \( 1 + 12.9iT - 71T^{2} \)
73 \( 1 - 8.63iT - 73T^{2} \)
79 \( 1 + 13.8iT - 79T^{2} \)
83 \( 1 + (-7.95 + 7.95i)T - 83iT^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 - 4.55T + 97T^{2} \)
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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

−11.72531699838810701437531671094, −10.42685698449122393023462321065, −9.432568021938645920802031075614, −9.111582485646285476299382791449, −7.20209374702313886245453435227, −6.81853490806442223896532000480, −5.03228691580670597060185499778, −4.17569413789645157772136952706, −2.94596221956239586963221756759, −1.85049465269201974588259815943, 2.44532899536332371499184392052, 3.50962024336141247988388726470, 4.51955135097874497524039703723, 5.92281788801028149503912564152, 6.84177332315141347137478548292, 8.036790744199247102358206169231, 8.458474014375969302708674426350, 9.805271461444573790825102107020, 10.68587484056800598784281103207, 12.33628781230324318190673978236

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