↑

Properties

Label 2-336-16.5-c1-0-16
Degree $2$
Conductor $336$
Sign $0.0599 + 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.647 − 1.25i)2-s + (0.707 − 0.707i)3-s + (−1.16 − 1.62i)4-s + (2.52 + 2.52i)5-s + (−0.430 − 1.34i)6-s − i·7-s + (−2.79 + 0.403i)8-s − 1.00i·9-s + (4.80 − 1.53i)10-s + (−1.92 − 1.92i)11-s + (−1.97 − 0.331i)12-s + (4.58 − 4.58i)13-s + (−1.25 − 0.647i)14-s + 3.56·15-s + (−1.30 + 3.78i)16-s + 3.46·17-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (0.408 − 0.408i)3-s + (−0.580 − 0.814i)4-s + (1.12 + 1.12i)5-s + (−0.175 − 0.549i)6-s − 0.377i·7-s + (−0.989 + 0.142i)8-s − 0.333i·9-s + (1.51 − 0.485i)10-s + (−0.580 − 0.580i)11-s + (−0.569 − 0.0956i)12-s + (1.27 − 1.27i)13-s + (−0.335 − 0.173i)14-s + 0.920·15-s + (−0.326 + 0.945i)16-s + 0.840·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0599 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0599 + 0.998i$
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: \((2,\ 336,\ (\ :1/2),\ 0.0599 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48252 - 1.39620i\)
\(L(\frac12)\) \(\approx\) \(1.48252 - 1.39620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.647 + 1.25i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-2.52 - 2.52i)T + 5iT^{2} \)
11 \( 1 + (1.92 + 1.92i)T + 11iT^{2} \)
13 \( 1 + (-4.58 + 4.58i)T - 13iT^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + (3.90 - 3.90i)T - 19iT^{2} \)
23 \( 1 - 2.95iT - 23T^{2} \)
29 \( 1 + (1.67 - 1.67i)T - 29iT^{2} \)
31 \( 1 + 7.77T + 31T^{2} \)
37 \( 1 + (3.39 + 3.39i)T + 37iT^{2} \)
41 \( 1 - 8.22iT - 41T^{2} \)
43 \( 1 + (-3.36 - 3.36i)T + 43iT^{2} \)
47 \( 1 - 5.44T + 47T^{2} \)
53 \( 1 + (-0.328 - 0.328i)T + 53iT^{2} \)
59 \( 1 + (-4.92 - 4.92i)T + 59iT^{2} \)
61 \( 1 + (10.0 - 10.0i)T - 61iT^{2} \)
67 \( 1 + (5.72 - 5.72i)T - 67iT^{2} \)
71 \( 1 + 5.86iT - 71T^{2} \)
73 \( 1 + 6.93iT - 73T^{2} \)
79 \( 1 - 8.37T + 79T^{2} \)
83 \( 1 + (-3.79 + 3.79i)T - 83iT^{2} \)
89 \( 1 + 7.71iT - 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.96478625237840439013902691682, −10.63547764371627607072651063616, −9.840907888015187987477384084482, −8.682696352130215926102002391782, −7.52268443294204338989320188272, −6.02083727926857049494399528560, −5.70320400324429594168577552036, −3.62493065001143499639921119311, −2.90033398798048310491375924282, −1.53377442871357088268151213795, 2.12908352369080207803620936917, 3.94095632991276000735864055264, 4.96132550246109443794636534997, 5.73886185691820447541061515296, 6.80069777880332423594771845259, 8.195076200613089492956695400983, 9.029312217843223549834620824423, 9.369056507150606774151163682765, 10.72761817145933475455781454866, 12.19847222463927076612341252167

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