↑

Properties

Label 2-336-16.5-c1-0-11
Degree $2$
Conductor $336$
Sign $0.851 - 0.524i$
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  + (1.41 − 0.0564i)2-s + (−0.707 + 0.707i)3-s + (1.99 − 0.159i)4-s + (0.616 + 0.616i)5-s + (−0.959 + 1.03i)6-s + i·7-s + (2.80 − 0.338i)8-s − 1.00i·9-s + (0.906 + 0.836i)10-s + (0.853 + 0.853i)11-s + (−1.29 + 1.52i)12-s + (0.0155 − 0.0155i)13-s + (0.0564 + 1.41i)14-s − 0.872·15-s + (3.94 − 0.636i)16-s + 1.42·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0399i)2-s + (−0.408 + 0.408i)3-s + (0.996 − 0.0798i)4-s + (0.275 + 0.275i)5-s + (−0.391 + 0.424i)6-s + 0.377i·7-s + (0.992 − 0.119i)8-s − 0.333i·9-s + (0.286 + 0.264i)10-s + (0.257 + 0.257i)11-s + (−0.374 + 0.439i)12-s + (0.00429 − 0.00429i)13-s + (0.0150 + 0.377i)14-s − 0.225·15-s + (0.987 − 0.159i)16-s + 0.345·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.851 - 0.524i$
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: \((2,\ 336,\ (\ :1/2),\ 0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16453 + 0.613623i\)
\(L(\frac12)\) \(\approx\) \(2.16453 + 0.613623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0564i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 - iT \)
good5 \( 1 + (-0.616 - 0.616i)T + 5iT^{2} \)
11 \( 1 + (-0.853 - 0.853i)T + 11iT^{2} \)
13 \( 1 + (-0.0155 + 0.0155i)T - 13iT^{2} \)
17 \( 1 - 1.42T + 17T^{2} \)
19 \( 1 + (3.18 - 3.18i)T - 19iT^{2} \)
23 \( 1 + 1.01iT - 23T^{2} \)
29 \( 1 + (-2.03 + 2.03i)T - 29iT^{2} \)
31 \( 1 + 5.35T + 31T^{2} \)
37 \( 1 + (3.88 + 3.88i)T + 37iT^{2} \)
41 \( 1 + 7.81iT - 41T^{2} \)
43 \( 1 + (1.36 + 1.36i)T + 43iT^{2} \)
47 \( 1 + 4.75T + 47T^{2} \)
53 \( 1 + (-2.45 - 2.45i)T + 53iT^{2} \)
59 \( 1 + (0.133 + 0.133i)T + 59iT^{2} \)
61 \( 1 + (7.49 - 7.49i)T - 61iT^{2} \)
67 \( 1 + (5.76 - 5.76i)T - 67iT^{2} \)
71 \( 1 + 9.60iT - 71T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + 8.34T + 79T^{2} \)
83 \( 1 + (-6.42 + 6.42i)T - 83iT^{2} \)
89 \( 1 - 3.25iT - 89T^{2} \)
97 \( 1 - 4.95T + 97T^{2} \)
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

−11.91439086302498705187694584865, −10.69554420424916632861494544690, −10.22428297655029419174835079459, −8.925044841880335900621136656774, −7.56574381913881704002930070645, −6.42288792791844519790557055303, −5.72538584895098833815766746469, −4.61673132079091156408774448738, −3.54200794355780461623561807330, −2.08874307305412467159998135181, 1.57699224816045814618366530492, 3.21103575004364666103313674385, 4.56265972780712448976749296120, 5.50592311788248488706105539490, 6.51100742393370213455890977531, 7.30362806728624057942906260979, 8.467259534091037218862800608346, 9.841235840902622030812299505470, 10.94446078125325116709936932043, 11.52549332786465932514805445884

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