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Properties

Label 2-336-112.19-c1-0-10
Degree $2$
Conductor $336$
Sign $0.996 + 0.0830i$
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  + (−1.09 + 0.895i)2-s + (0.258 − 0.965i)3-s + (0.394 − 1.96i)4-s + (−2.90 + 0.778i)5-s + (0.582 + 1.28i)6-s + (2.58 + 0.560i)7-s + (1.32 + 2.49i)8-s + (−0.866 − 0.499i)9-s + (2.48 − 3.45i)10-s + (−0.213 + 0.796i)11-s + (−1.79 − 0.888i)12-s + (1.43 − 1.43i)13-s + (−3.33 + 1.70i)14-s + 3.00i·15-s + (−3.68 − 1.54i)16-s + (6.92 − 3.99i)17-s + ⋯
L(s)  = 1  + (−0.773 + 0.633i)2-s + (0.149 − 0.557i)3-s + (0.197 − 0.980i)4-s + (−1.29 + 0.348i)5-s + (0.237 + 0.526i)6-s + (0.977 + 0.211i)7-s + (0.468 + 0.883i)8-s + (−0.288 − 0.166i)9-s + (0.784 − 1.09i)10-s + (−0.0643 + 0.240i)11-s + (−0.517 − 0.256i)12-s + (0.398 − 0.398i)13-s + (−0.890 + 0.455i)14-s + 0.776i·15-s + (−0.922 − 0.387i)16-s + (1.67 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.996 + 0.0830i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.996 + 0.0830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879819 - 0.0365983i\)
\(L(\frac12)\) \(\approx\) \(0.879819 - 0.0365983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 - 0.895i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-2.58 - 0.560i)T \)
good5 \( 1 + (2.90 - 0.778i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.213 - 0.796i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.43 + 1.43i)T - 13iT^{2} \)
17 \( 1 + (-6.92 + 3.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 + 1.37i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.40 + 4.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.35 + 3.35i)T + 29iT^{2} \)
31 \( 1 + (-0.102 - 0.176i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.86 - 6.94i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (5.29 + 5.29i)T + 43iT^{2} \)
47 \( 1 + (5.93 - 10.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.14 - 1.11i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.574 - 0.153i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.95 - 7.29i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (8.16 + 2.18i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 + (6.59 + 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.734 + 0.423i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.65 - 2.65i)T + 83iT^{2} \)
89 \( 1 + (-6.31 + 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.83iT - 97T^{2} \)
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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

−11.58292544672428888596404673468, −10.66988032064881289451784749420, −9.480215173653682707769934570053, −8.371165327432340856984875155468, −7.63020208531595963835206249491, −7.32027779403226308562728726608, −5.83440289236967329956002099270, −4.73073234255172549382613426798, −2.96490695838180932848185865468, −1.01815935338362682297505847025, 1.29420082778130298610000923479, 3.40902937061794866180927268039, 4.04183313658022257424657646652, 5.39529036744299629126048526492, 7.44925220990498037992105589993, 7.922242351183592074589528001526, 8.723912877102155351462137437666, 9.736531365796500281004018991551, 10.76828593137562476943033771281, 11.48036285953093843882348144940

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