↑

Properties

Label 2-336-112.101-c2-0-19
Degree $2$
Conductor $336$
Sign $0.726 - 0.687i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.99 − 0.103i)2-s + (−1.67 + 0.448i)3-s + (3.97 − 0.412i)4-s + (−3.88 − 1.04i)5-s + (−3.29 + 1.06i)6-s + (−3.30 + 6.17i)7-s + (7.90 − 1.23i)8-s + (2.59 − 1.50i)9-s + (−7.87 − 1.67i)10-s + (11.0 − 2.95i)11-s + (−6.47 + 2.47i)12-s + (15.1 + 15.1i)13-s + (−5.96 + 12.6i)14-s + 6.97·15-s + (15.6 − 3.28i)16-s + (20.9 + 12.1i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0516i)2-s + (−0.557 + 0.149i)3-s + (0.994 − 0.103i)4-s + (−0.777 − 0.208i)5-s + (−0.549 + 0.178i)6-s + (−0.472 + 0.881i)7-s + (0.988 − 0.154i)8-s + (0.288 − 0.166i)9-s + (−0.787 − 0.167i)10-s + (1.00 − 0.268i)11-s + (−0.539 + 0.206i)12-s + (1.16 + 1.16i)13-s + (−0.425 + 0.904i)14-s + 0.464·15-s + (0.978 − 0.205i)16-s + (1.23 + 0.712i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.726 - 0.687i$
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: \((2,\ 336,\ (\ :1),\ 0.726 - 0.687i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23334 + 0.889642i\)
\(L(\frac12)\) \(\approx\) \(2.23334 + 0.889642i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.103i)T \)
3 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 + (3.30 - 6.17i)T \)
good5 \( 1 + (3.88 + 1.04i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-11.0 + 2.95i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-15.1 - 15.1i)T + 169iT^{2} \)
17 \( 1 + (-20.9 - 12.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.76 - 25.2i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-7.31 + 4.22i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (18.5 - 18.5i)T - 841iT^{2} \)
31 \( 1 + (40.8 + 23.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.6 + 54.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 51.7T + 1.68e3T^{2} \)
43 \( 1 + (-10.1 - 10.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (34.3 - 19.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (5.21 - 1.39i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-7.49 - 27.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-7.00 + 26.1i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-21.8 - 81.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 13.8iT - 5.04e3T^{2} \)
73 \( 1 + (-14.9 + 25.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-18.2 - 31.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (15.8 + 15.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (82.7 + 143. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 190. iT - 9.40e3T^{2} \)
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

−11.51919445142757007695674329222, −11.06238570964990302958125993481, −9.698739763557414835918250179599, −8.609352848848261835542484185410, −7.38930196992131822524470701957, −6.09634085675266673047377379413, −5.79108573822394020013942003822, −4.08410531119026401184805394727, −3.64979030908357807406826806788, −1.61649788581167967041126025065, 0.995164479703314144827948055281, 3.25120305043041158224351453773, 3.97928531903873052202641491152, 5.22563577118630342675860874896, 6.35455398830619763357692752006, 7.16336769214554588806478887084, 7.919836442079049335964657649922, 9.599444569828212752394580859090, 10.81431314355913878257689494576, 11.26184440546153555619743639222

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