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Properties

Label 2-336-21.5-c3-0-45
Degree $2$
Conductor $336$
Sign $-0.933 - 0.359i$
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  + (−0.588 − 5.16i)3-s + (4.27 − 7.41i)5-s + (6.41 − 17.3i)7-s + (−26.3 + 6.07i)9-s + (−53.8 + 31.0i)11-s − 61.7i·13-s + (−40.7 − 17.7i)15-s + (−13.3 − 23.0i)17-s + (58.6 + 33.8i)19-s + (−93.4 − 22.9i)21-s + (−45.8 − 26.4i)23-s + (25.8 + 44.8i)25-s + (46.8 + 132. i)27-s − 55.1i·29-s + (134. − 77.4i)31-s + ⋯
L(s)  = 1  + (−0.113 − 0.993i)3-s + (0.382 − 0.662i)5-s + (0.346 − 0.938i)7-s + (−0.974 + 0.224i)9-s + (−1.47 + 0.852i)11-s − 1.31i·13-s + (−0.701 − 0.305i)15-s + (−0.190 − 0.329i)17-s + (0.707 + 0.408i)19-s + (−0.971 − 0.238i)21-s + (−0.415 − 0.239i)23-s + (0.207 + 0.358i)25-s + (0.333 + 0.942i)27-s − 0.352i·29-s + (0.776 − 0.448i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.933 - 0.359i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.933 - 0.359i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9543527561\)
\(L(\frac12)\) \(\approx\) \(0.9543527561\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.588 + 5.16i)T \)
7 \( 1 + (-6.41 + 17.3i)T \)
good5 \( 1 + (-4.27 + 7.41i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (53.8 - 31.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 61.7iT - 2.19e3T^{2} \)
17 \( 1 + (13.3 + 23.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-58.6 - 33.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (45.8 + 26.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 55.1iT - 2.43e4T^{2} \)
31 \( 1 + (-134. + 77.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (157. - 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 210.T + 6.89e4T^{2} \)
43 \( 1 + 351.T + 7.95e4T^{2} \)
47 \( 1 + (-115. + 200. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (232. - 134. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-9.14 - 15.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.3 + 41.7i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-64.7 - 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 804. iT - 3.57e5T^{2} \)
73 \( 1 + (-370. + 213. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-609. + 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.37e3T + 5.71e5T^{2} \)
89 \( 1 + (-386. + 670. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 848. iT - 9.12e5T^{2} \)
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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

−10.50175323290984669727656241347, −9.901372606101826555816523468959, −8.314029795927196504168727358119, −7.81706473983979954873601649808, −6.91420357784650130766041754680, −5.52510038121637702965570304175, −4.85971506118035040596258320225, −2.97942712239526592685649670595, −1.58726399200596560042701419118, −0.33009620337993555680713197546, 2.30071844857346588245077774581, 3.29767196270942280077791216025, 4.82189131312494387147014464388, 5.61452933479070401285683452258, 6.61120557186248997304392163833, 8.139236926679961567118668097393, 8.924818396193382978176483999230, 9.863787756149726954676539451886, 10.75843072026274426980171641953, 11.36357092288907673197937742502

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