↑

Properties

Label 2-336-112.101-c2-0-1
Degree $2$
Conductor $336$
Sign $-0.760 - 0.649i$
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  + (−1.77 − 0.918i)2-s + (1.67 − 0.448i)3-s + (2.31 + 3.26i)4-s + (−5.33 − 1.42i)5-s + (−3.38 − 0.740i)6-s + (1.11 + 6.91i)7-s + (−1.10 − 7.92i)8-s + (2.59 − 1.50i)9-s + (8.16 + 7.43i)10-s + (6.55 − 1.75i)11-s + (5.33 + 4.42i)12-s + (−10.4 − 10.4i)13-s + (4.37 − 13.2i)14-s − 9.56·15-s + (−5.31 + 15.0i)16-s + (−16.8 − 9.72i)17-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)2-s + (0.557 − 0.149i)3-s + (0.577 + 0.816i)4-s + (−1.06 − 0.285i)5-s + (−0.563 − 0.123i)6-s + (0.158 + 0.987i)7-s + (−0.138 − 0.990i)8-s + (0.288 − 0.166i)9-s + (0.816 + 0.743i)10-s + (0.595 − 0.159i)11-s + (0.444 + 0.368i)12-s + (−0.802 − 0.802i)13-s + (0.312 − 0.949i)14-s − 0.637·15-s + (−0.331 + 0.943i)16-s + (−0.990 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.760 - 0.649i$
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: \((2,\ 336,\ (\ :1),\ -0.760 - 0.649i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0474410 + 0.128689i\)
\(L(\frac12)\) \(\approx\) \(0.0474410 + 0.128689i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.77 + 0.918i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-1.11 - 6.91i)T \)
good5 \( 1 + (5.33 + 1.42i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-6.55 + 1.75i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (10.4 + 10.4i)T + 169iT^{2} \)
17 \( 1 + (16.8 + 9.72i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (8.59 - 32.0i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (6.24 - 3.60i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (12.3 - 12.3i)T - 841iT^{2} \)
31 \( 1 + (48.7 + 28.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-2.34 + 8.76i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 20.7T + 1.68e3T^{2} \)
43 \( 1 + (-52.8 - 52.8i)T + 1.84e3iT^{2} \)
47 \( 1 + (50.0 - 28.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (54.9 - 14.7i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-26.9 - 100. i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (16.9 - 63.3i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (7.05 + 26.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 99.8iT - 5.04e3T^{2} \)
73 \( 1 + (-58.7 + 101. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (8.93 + 15.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (95.6 + 95.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-24.5 - 42.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 22.5iT - 9.40e3T^{2} \)
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

−11.73264903444935935847983345334, −10.82666555294474741495082995977, −9.565710010547531973323351886200, −8.920748846888379100668546143626, −7.988302865283916387313290654886, −7.50429916305787880351288638392, −6.03729680790112297012222872212, −4.29003372968281660937357419309, −3.16931767505791564723238427409, −1.87605858092807234283731917237, 0.07523987299526948505462450118, 2.03946636356303715386704076809, 3.80534775995027929854216816297, 4.78407191172083048072280026551, 6.87906205605763768648298417964, 7.03723114732190768425219503231, 8.159895877910164001367731763998, 8.999705034655653793437068318360, 9.862806614711690636414516328610, 11.06493576760151596102959535740

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