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Properties

Label 2-336-112.101-c2-0-47
Degree $2$
Conductor $336$
Sign $0.895 - 0.446i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.86 + 0.715i)2-s + (1.67 − 0.448i)3-s + (2.97 + 2.67i)4-s + (4.29 + 1.14i)5-s + (3.44 + 0.359i)6-s + (−5.40 − 4.44i)7-s + (3.64 + 7.12i)8-s + (2.59 − 1.50i)9-s + (7.19 + 5.21i)10-s + (19.1 − 5.13i)11-s + (6.17 + 3.13i)12-s + (−12.5 − 12.5i)13-s + (−6.91 − 12.1i)14-s + 7.69·15-s + (1.71 + 15.9i)16-s + (13.6 + 7.86i)17-s + ⋯
L(s)  = 1  + (0.933 + 0.357i)2-s + (0.557 − 0.149i)3-s + (0.744 + 0.668i)4-s + (0.858 + 0.229i)5-s + (0.574 + 0.0599i)6-s + (−0.772 − 0.635i)7-s + (0.455 + 0.890i)8-s + (0.288 − 0.166i)9-s + (0.719 + 0.521i)10-s + (1.74 − 0.466i)11-s + (0.514 + 0.261i)12-s + (−0.966 − 0.966i)13-s + (−0.494 − 0.869i)14-s + 0.513·15-s + (0.107 + 0.994i)16-s + (0.801 + 0.462i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.446i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.895 - 0.446i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.895 - 0.446i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.73614 + 0.879379i\)
\(L(\frac12)\) \(\approx\) \(3.73614 + 0.879379i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.86 - 0.715i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (5.40 + 4.44i)T \)
good5 \( 1 + (-4.29 - 1.14i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-19.1 + 5.13i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (12.5 + 12.5i)T + 169iT^{2} \)
17 \( 1 + (-13.6 - 7.86i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (7.64 - 28.5i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (32.0 - 18.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-22.9 + 22.9i)T - 841iT^{2} \)
31 \( 1 + (6.11 + 3.53i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-6.62 + 24.7i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 44.3T + 1.68e3T^{2} \)
43 \( 1 + (19.5 + 19.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (37.1 - 21.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-47.0 + 12.6i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-2.47 - 9.21i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-6.51 + 24.3i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (8.58 + 32.0i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 22.5iT - 5.04e3T^{2} \)
73 \( 1 + (26.7 - 46.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (11.8 + 20.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (82.2 + 82.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (-7.79 - 13.5i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 102. iT - 9.40e3T^{2} \)
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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

−11.81410422893012272412364741289, −10.19297012788271917395255832902, −9.836918411731490098047689474829, −8.332175848445954904307181543029, −7.42746485671537351689052191534, −6.30422587535186131001494251801, −5.81680309239540242467952489057, −4.02454333859744370065456758817, −3.32219393941039871616873843509, −1.80977203958813710697903174394, 1.74568337051146560212096550686, 2.77607167798607245624238574724, 4.11627768160260473535978639727, 5.11614188306747980472433468574, 6.44917228189201691879906278350, 6.91631766528711629060985368502, 8.844454133972139024170563573291, 9.618781713219420731067201124278, 10.05796233103195608229778183915, 11.73780505792126587253602678148

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