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Properties

Label 2-336-48.11-c1-0-28
Degree $2$
Conductor $336$
Sign $0.970 - 0.239i$
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  + (1.20 + 0.742i)2-s + (−1.65 − 0.505i)3-s + (0.897 + 1.78i)4-s + (2.00 − 2.00i)5-s + (−1.61 − 1.83i)6-s + 7-s + (−0.246 + 2.81i)8-s + (2.48 + 1.67i)9-s + (3.90 − 0.925i)10-s + (−2.67 − 2.67i)11-s + (−0.583 − 3.41i)12-s + (4.14 − 4.14i)13-s + (1.20 + 0.742i)14-s + (−4.34 + 2.31i)15-s + (−2.38 + 3.20i)16-s + 3.45i·17-s + ⋯
L(s)  = 1  + (0.851 + 0.525i)2-s + (−0.956 − 0.291i)3-s + (0.448 + 0.893i)4-s + (0.897 − 0.897i)5-s + (−0.660 − 0.750i)6-s + 0.377·7-s + (−0.0872 + 0.996i)8-s + (0.829 + 0.558i)9-s + (1.23 − 0.292i)10-s + (−0.806 − 0.806i)11-s + (−0.168 − 0.985i)12-s + (1.14 − 1.14i)13-s + (0.321 + 0.198i)14-s + (−1.12 + 0.596i)15-s + (−0.597 + 0.802i)16-s + 0.838i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.970 - 0.239i$
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: \((2,\ 336,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89398 + 0.230111i\)
\(L(\frac12)\) \(\approx\) \(1.89398 + 0.230111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 0.742i)T \)
3 \( 1 + (1.65 + 0.505i)T \)
7 \( 1 - T \)
good5 \( 1 + (-2.00 + 2.00i)T - 5iT^{2} \)
11 \( 1 + (2.67 + 2.67i)T + 11iT^{2} \)
13 \( 1 + (-4.14 + 4.14i)T - 13iT^{2} \)
17 \( 1 - 3.45iT - 17T^{2} \)
19 \( 1 + (-5.27 - 5.27i)T + 19iT^{2} \)
23 \( 1 - 2.11iT - 23T^{2} \)
29 \( 1 + (0.808 + 0.808i)T + 29iT^{2} \)
31 \( 1 + 7.56iT - 31T^{2} \)
37 \( 1 + (1.06 + 1.06i)T + 37iT^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + (5.32 - 5.32i)T - 43iT^{2} \)
47 \( 1 + 6.19T + 47T^{2} \)
53 \( 1 + (-0.414 + 0.414i)T - 53iT^{2} \)
59 \( 1 + (7.26 + 7.26i)T + 59iT^{2} \)
61 \( 1 + (1.06 - 1.06i)T - 61iT^{2} \)
67 \( 1 + (-4.81 - 4.81i)T + 67iT^{2} \)
71 \( 1 - 1.83iT - 71T^{2} \)
73 \( 1 - 0.150iT - 73T^{2} \)
79 \( 1 - 4.73iT - 79T^{2} \)
83 \( 1 + (-1.94 + 1.94i)T - 83iT^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−11.74282290808199147167401570545, −10.92662728286887544021543587085, −9.930650374995689552189600946001, −8.314527300051043530747238394109, −7.83827565151299046434847192669, −6.21707409699348035180916435130, −5.62762918185734628765940151725, −5.11531175720489199336286818607, −3.54809207697335053099821007009, −1.53868203529529638490652471365, 1.72139882380721138613988842042, 3.16974787431136057885574494202, 4.71551502275804338069204592115, 5.34644458606740129957974500716, 6.60729942634610144336202658689, 7.00428360808134562928268191215, 9.214180344948946160776321251442, 10.08972341504892819259260027423, 10.74997781108764188683321355911, 11.47762598945586543216900251792

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