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Properties

Label 2-336-112.101-c2-0-30
Degree $2$
Conductor $336$
Sign $0.387 - 0.921i$
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  + (0.426 + 1.95i)2-s + (1.67 − 0.448i)3-s + (−3.63 + 1.66i)4-s + (2.05 + 0.551i)5-s + (1.58 + 3.07i)6-s + (4.66 − 5.21i)7-s + (−4.80 − 6.39i)8-s + (2.59 − 1.50i)9-s + (−0.200 + 4.25i)10-s + (4.17 − 1.11i)11-s + (−5.33 + 4.41i)12-s + (11.4 + 11.4i)13-s + (12.1 + 6.89i)14-s + 3.68·15-s + (10.4 − 12.1i)16-s + (19.0 + 10.9i)17-s + ⋯
L(s)  = 1  + (0.213 + 0.977i)2-s + (0.557 − 0.149i)3-s + (−0.909 + 0.416i)4-s + (0.411 + 0.110i)5-s + (0.264 + 0.513i)6-s + (0.666 − 0.745i)7-s + (−0.600 − 0.799i)8-s + (0.288 − 0.166i)9-s + (−0.0200 + 0.425i)10-s + (0.379 − 0.101i)11-s + (−0.444 + 0.368i)12-s + (0.877 + 0.877i)13-s + (0.870 + 0.492i)14-s + 0.245·15-s + (0.653 − 0.757i)16-s + (1.11 + 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.387 - 0.921i$
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: \((2,\ 336,\ (\ :1),\ 0.387 - 0.921i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.01492 + 1.33801i\)
\(L(\frac12)\) \(\approx\) \(2.01492 + 1.33801i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.426 - 1.95i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-4.66 + 5.21i)T \)
good5 \( 1 + (-2.05 - 0.551i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-4.17 + 1.11i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (-11.4 - 11.4i)T + 169iT^{2} \)
17 \( 1 + (-19.0 - 10.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.06 - 11.4i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-0.432 + 0.249i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-13.1 + 13.1i)T - 841iT^{2} \)
31 \( 1 + (16.1 + 9.32i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (12.4 - 46.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 5.75T + 1.68e3T^{2} \)
43 \( 1 + (-15.6 - 15.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (-12.8 + 7.41i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (29.4 - 7.89i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (10.1 + 37.7i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (2.03 - 7.59i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (20.9 + 78.3i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 86.4iT - 5.04e3T^{2} \)
73 \( 1 + (46.2 - 80.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-76.0 - 131. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (34.1 + 34.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (72.5 + 125. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 26.6iT - 9.40e3T^{2} \)
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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

−11.63970506747517942089682231236, −10.33277362723386477660546764967, −9.465171508357960517205917934379, −8.379150112053978144332115199399, −7.80204365912926049795212935304, −6.67342375662973231698770546386, −5.84723927492726622546182828335, −4.40786479768164331254438280178, −3.56592630584128836252874006080, −1.46463947747738082629202209137, 1.32125696377404032521879373121, 2.60801607793882614785467146359, 3.71769010611264203256206649289, 5.08024345364949126988021425549, 5.83599677938421053510147861124, 7.66298113886843473671221162420, 8.730106504431432274345223589170, 9.281182282312862202250471284421, 10.31300411682511401194607961824, 11.15410861361318693384845041209

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