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Properties

Label 2-336-112.101-c2-0-53
Degree $2$
Conductor $336$
Sign $-0.480 + 0.876i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 1.47i)2-s + (1.67 − 0.448i)3-s + (−0.331 − 3.98i)4-s + (−2.13 − 0.572i)5-s + (−1.60 + 3.06i)6-s + (0.973 − 6.93i)7-s + (6.31 + 4.91i)8-s + (2.59 − 1.50i)9-s + (3.73 − 2.36i)10-s + (−13.3 + 3.57i)11-s + (−2.34 − 6.52i)12-s + (2.55 + 2.55i)13-s + (8.88 + 10.8i)14-s − 3.83·15-s + (−15.7 + 2.64i)16-s + (−12.7 − 7.36i)17-s + ⋯
L(s)  = 1  + (−0.677 + 0.735i)2-s + (0.557 − 0.149i)3-s + (−0.0828 − 0.996i)4-s + (−0.427 − 0.114i)5-s + (−0.267 + 0.511i)6-s + (0.139 − 0.990i)7-s + (0.789 + 0.613i)8-s + (0.288 − 0.166i)9-s + (0.373 − 0.236i)10-s + (−1.21 + 0.324i)11-s + (−0.195 − 0.543i)12-s + (0.196 + 0.196i)13-s + (0.634 + 0.772i)14-s − 0.255·15-s + (−0.986 + 0.165i)16-s + (−0.750 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.480 + 0.876i$
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: \((2,\ 336,\ (\ :1),\ -0.480 + 0.876i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.255387 - 0.431161i\)
\(L(\frac12)\) \(\approx\) \(0.255387 - 0.431161i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 1.47i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-0.973 + 6.93i)T \)
good5 \( 1 + (2.13 + 0.572i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (13.3 - 3.57i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-2.55 - 2.55i)T + 169iT^{2} \)
17 \( 1 + (12.7 + 7.36i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.95 + 7.28i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (27.0 - 15.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-5.66 + 5.66i)T - 841iT^{2} \)
31 \( 1 + (24.6 + 14.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.40 + 5.22i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 24.1T + 1.68e3T^{2} \)
43 \( 1 + (-2.97 - 2.97i)T + 1.84e3iT^{2} \)
47 \( 1 + (-20.0 + 11.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-56.6 + 15.1i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (13.0 + 48.8i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (1.16 - 4.34i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (27.6 + 103. i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 51.0iT - 5.04e3T^{2} \)
73 \( 1 + (-51.5 + 89.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (24.5 + 42.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (100. + 100. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-77.0 - 133. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 125. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.73221770975187552739417232080, −9.983130949920724707939810805005, −9.033982527282665675566906707419, −7.86996271980728925474641449344, −7.59483287560306271772684662036, −6.48850337583700191999670280699, −5.07941141138355387948851576760, −3.97834381559031610905426873960, −2.07451432834343292335907126629, −0.25566328885759167805063344205, 2.02737325643763724219213708817, 3.03289576099256711458015759514, 4.22609398631895701752705820016, 5.69989845895257519368036747046, 7.29783544263922119059715827035, 8.305430623232816137969408468787, 8.652118035431154489153776164887, 9.862144045964786074412436540376, 10.63830477060336814350397153340, 11.51417637137573434108256769618

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