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Properties

Label 2-336-12.11-c3-0-4
Degree $2$
Conductor $336$
Sign $-0.907 + 0.420i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.18 + 4.71i)3-s + 14.5i·5-s − 7i·7-s + (−17.4 + 20.6i)9-s − 31.0·11-s − 55.7·13-s + (−68.7 + 31.8i)15-s − 86.4i·17-s + 96.8i·19-s + (32.9 − 15.3i)21-s + 115.·23-s − 87.7·25-s + (−135. − 37.0i)27-s − 49.1i·29-s − 12.5i·31-s + ⋯
L(s)  = 1  + (0.420 + 0.907i)3-s + 1.30i·5-s − 0.377i·7-s + (−0.645 + 0.763i)9-s − 0.850·11-s − 1.18·13-s + (−1.18 + 0.549i)15-s − 1.23i·17-s + 1.16i·19-s + (0.342 − 0.159i)21-s + 1.04·23-s − 0.701·25-s + (−0.964 − 0.264i)27-s − 0.314i·29-s − 0.0728i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8652905479\)
\(L(\frac12)\) \(\approx\) \(0.8652905479\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-2.18 - 4.71i)T \)
7 \( 1 + 7iT \)
good5 \( 1 - 14.5iT - 125T^{2} \)
11 \( 1 + 31.0T + 1.33e3T^{2} \)
13 \( 1 + 55.7T + 2.19e3T^{2} \)
17 \( 1 + 86.4iT - 4.91e3T^{2} \)
19 \( 1 - 96.8iT - 6.85e3T^{2} \)
23 \( 1 - 115.T + 1.21e4T^{2} \)
29 \( 1 + 49.1iT - 2.43e4T^{2} \)
31 \( 1 + 12.5iT - 2.97e4T^{2} \)
37 \( 1 + 296.T + 5.06e4T^{2} \)
41 \( 1 + 213. iT - 6.89e4T^{2} \)
43 \( 1 - 165. iT - 7.95e4T^{2} \)
47 \( 1 - 426.T + 1.03e5T^{2} \)
53 \( 1 - 460. iT - 1.48e5T^{2} \)
59 \( 1 + 686.T + 2.05e5T^{2} \)
61 \( 1 + 583.T + 2.26e5T^{2} \)
67 \( 1 - 589. iT - 3.00e5T^{2} \)
71 \( 1 - 766.T + 3.57e5T^{2} \)
73 \( 1 + 904.T + 3.89e5T^{2} \)
79 \( 1 - 459. iT - 4.93e5T^{2} \)
83 \( 1 + 1.11e3T + 5.71e5T^{2} \)
89 \( 1 - 119. iT - 7.04e5T^{2} \)
97 \( 1 + 331.T + 9.12e5T^{2} \)
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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

−11.33273375303380445854597305572, −10.49853900037268939084902666339, −10.05960835111229172960668626100, −9.049117468845695478652752812919, −7.67658069744261758686704633269, −7.14695277747467357174753921196, −5.62781066156865040494831897748, −4.58106673424733767950532038568, −3.23348519471406350390522244835, −2.50911486267979513643280238655, 0.27293341319712162881370459328, 1.70996419128607998155574944977, 2.94749932909651930605157095015, 4.71028954839488219270176059885, 5.54222347150205403428432620229, 6.88001144716898422615624887111, 7.84971019953567437845796093464, 8.709339675023603240243672020173, 9.283917423285647442921535570069, 10.62297495608501346717467098320

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