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Properties

Label 2-336-16.5-c1-0-3
Degree $2$
Conductor $336$
Sign $-0.179 - 0.983i$
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  + (−1.33 + 0.470i)2-s + (−0.707 + 0.707i)3-s + (1.55 − 1.25i)4-s + (2.01 + 2.01i)5-s + (0.609 − 1.27i)6-s + i·7-s + (−1.48 + 2.40i)8-s − 1.00i·9-s + (−3.62 − 1.73i)10-s + (1.19 + 1.19i)11-s + (−0.212 + 1.98i)12-s + (1.92 − 1.92i)13-s + (−0.470 − 1.33i)14-s − 2.84·15-s + (0.844 − 3.90i)16-s + 1.24·17-s + ⋯
L(s)  = 1  + (−0.942 + 0.333i)2-s + (−0.408 + 0.408i)3-s + (0.778 − 0.628i)4-s + (0.898 + 0.898i)5-s + (0.248 − 0.520i)6-s + 0.377i·7-s + (−0.524 + 0.851i)8-s − 0.333i·9-s + (−1.14 − 0.548i)10-s + (0.361 + 0.361i)11-s + (−0.0612 + 0.574i)12-s + (0.533 − 0.533i)13-s + (−0.125 − 0.356i)14-s − 0.733·15-s + (0.211 − 0.977i)16-s + 0.303·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.179 - 0.983i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.179 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561176 + 0.672502i\)
\(L(\frac12)\) \(\approx\) \(0.561176 + 0.672502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.470i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (-2.01 - 2.01i)T + 5iT^{2} \)
11 \( 1 + (-1.19 - 1.19i)T + 11iT^{2} \)
13 \( 1 + (-1.92 + 1.92i)T - 13iT^{2} \)
17 \( 1 - 1.24T + 17T^{2} \)
19 \( 1 + (0.720 - 0.720i)T - 19iT^{2} \)
23 \( 1 - 8.18iT - 23T^{2} \)
29 \( 1 + (1.27 - 1.27i)T - 29iT^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (-5.38 - 5.38i)T + 37iT^{2} \)
41 \( 1 - 4.30iT - 41T^{2} \)
43 \( 1 + (-0.310 - 0.310i)T + 43iT^{2} \)
47 \( 1 - 6.10T + 47T^{2} \)
53 \( 1 + (1.34 + 1.34i)T + 53iT^{2} \)
59 \( 1 + (10.2 + 10.2i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + (-1.99 + 1.99i)T - 67iT^{2} \)
71 \( 1 + 4.97iT - 71T^{2} \)
73 \( 1 - 9.75iT - 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + (-3.53 + 3.53i)T - 83iT^{2} \)
89 \( 1 - 2.86iT - 89T^{2} \)
97 \( 1 + 8.40T + 97T^{2} \)
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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

−11.28501473380589201094730378420, −10.83257571565127363113144531235, −9.722199111927443722509996161623, −9.426178347359457695937876961937, −8.053608155333098322064101556232, −6.96535227096746456726874667993, −6.05404308024399428748159187949, −5.37224668928978372403134585446, −3.30574742095680225244416692924, −1.78436902924377403352887631387, 0.932689144796110782966526587845, 2.16828113426192447865831003835, 4.05270387264075577233770475733, 5.63860163734182200980195149406, 6.53113023569836459089400691648, 7.59030054391897883658256673576, 8.804660840134332152407745142441, 9.235508556284778154566029816801, 10.44576963324474063188144186978, 11.09677454996723019531489548469

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