Properties

Label 2-336-28.3-c1-0-1
Degree $2$
Conductor $336$
Sign $0.667 - 0.744i$
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  + (−0.5 + 0.866i)3-s + (1.5 − 0.866i)5-s + (0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (1.5 + 0.866i)11-s + 1.73i·15-s + (3 + 1.73i)17-s + (1 + 1.73i)19-s + (−2.5 − 0.866i)21-s + (−1 + 1.73i)25-s + 0.999·27-s + 9·29-s + (−2.5 + 4.33i)31-s + (−1.5 + 0.866i)33-s + (3 + 3.46i)35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 − 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (0.452 + 0.261i)11-s + 0.447i·15-s + (0.727 + 0.420i)17-s + (0.229 + 0.397i)19-s + (−0.545 − 0.188i)21-s + (−0.200 + 0.346i)25-s + 0.192·27-s + 1.67·29-s + (−0.449 + 0.777i)31-s + (−0.261 + 0.150i)33-s + (0.507 + 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.667 - 0.744i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26437 + 0.564994i\)
\(L(\frac12)\) \(\approx\) \(1.26437 + 0.564994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (12 + 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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

−11.86955287888147306674300484446, −10.63515159240329913830380747356, −9.776979601332275083691057184293, −9.031441912788082727672425083054, −8.138714065745209770547273243674, −6.62133921340766585861987217264, −5.63013305515793625995455845003, −4.91742076453252468906770468423, −3.41515829207316476287489313638, −1.76568093759239594602061038249, 1.19226971877493348484139312883, 2.86885285692459854617263203460, 4.38546512750434789043718867075, 5.68545939867883456956698803485, 6.65753218298887524195105826938, 7.42819210345826005041097765573, 8.531191304456363468334837613519, 9.819315281793336142955598128581, 10.46377590517617138320775756959, 11.44745381110301718548587896288

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