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Properties

Label 2-336-7.3-c2-0-15
Degree $2$
Conductor $336$
Sign $-0.994 + 0.107i$
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  + (−1.5 − 0.866i)3-s + (−4.68 + 2.70i)5-s + (6.12 − 3.38i)7-s + (1.5 + 2.59i)9-s + (5.26 − 9.12i)11-s + 12.0i·13-s + 9.36·15-s + (−20.8 − 12.0i)17-s + (−30.9 + 17.8i)19-s + (−12.1 − 0.234i)21-s + (−20.6 − 35.7i)23-s + (2.11 − 3.65i)25-s − 5.19i·27-s − 28.6·29-s + (4.50 + 2.59i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.936 + 0.540i)5-s + (0.875 − 0.483i)7-s + (0.166 + 0.288i)9-s + (0.478 − 0.829i)11-s + 0.925i·13-s + 0.624·15-s + (−1.22 − 0.707i)17-s + (−1.62 + 0.939i)19-s + (−0.577 − 0.0111i)21-s + (−0.898 − 1.55i)23-s + (0.0844 − 0.146i)25-s − 0.192i·27-s − 0.988·29-s + (0.145 + 0.0838i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.994 + 0.107i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ -0.994 + 0.107i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00857731 - 0.159559i\)
\(L(\frac12)\) \(\approx\) \(0.00857731 - 0.159559i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-6.12 + 3.38i)T \)
good5 \( 1 + (4.68 - 2.70i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-5.26 + 9.12i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 12.0iT - 169T^{2} \)
17 \( 1 + (20.8 + 12.0i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (30.9 - 17.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (20.6 + 35.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 28.6T + 841T^{2} \)
31 \( 1 + (-4.50 - 2.59i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-1.90 - 3.30i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 9.20iT - 1.68e3T^{2} \)
43 \( 1 + 54.2T + 1.84e3T^{2} \)
47 \( 1 + (14.3 - 8.27i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (9.58 - 16.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (10.8 + 6.25i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-84.1 + 48.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-0.499 + 0.864i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 14.8T + 5.04e3T^{2} \)
73 \( 1 + (51.0 + 29.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (50.7 + 87.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 22.7iT - 6.88e3T^{2} \)
89 \( 1 + (98.7 - 57.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 153. iT - 9.40e3T^{2} \)
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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

−11.26905592559410277269426912752, −10.29701561948734363421882232408, −8.741858924040931877498439544503, −8.035581502734303958826963357897, −6.92743829838342126955057955880, −6.25605826211344932773270706088, −4.60698008459777101656523802996, −3.88369552922730491464105663728, −1.98704225440093496642361546255, −0.07634347305207055111978536205, 1.91692370663978120010813913049, 3.97910966239441421056461226038, 4.66067818763206964525171206115, 5.76976007686696100259020006065, 7.04974653545930704254027162744, 8.166063446581179188732861909978, 8.808646188358000764759908707905, 10.01588909962827357887816594222, 11.15012245860845834645727037107, 11.60191858047015168162467675013

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