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Properties

Label 2-336-21.5-c3-0-3
Degree $2$
Conductor $336$
Sign $-0.474 + 0.880i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.12 + 0.843i)3-s + (−8.29 + 14.3i)5-s + (−9.36 + 15.9i)7-s + (25.5 − 8.65i)9-s + (−36.0 + 20.8i)11-s + 71.9i·13-s + (30.4 − 80.6i)15-s + (46.0 + 79.8i)17-s + (−86.6 − 50.0i)19-s + (34.5 − 89.8i)21-s + (74.8 + 43.2i)23-s + (−75.1 − 130. i)25-s + (−123. + 65.9i)27-s + 45.0i·29-s + (110. − 63.8i)31-s + ⋯
L(s)  = 1  + (−0.986 + 0.162i)3-s + (−0.742 + 1.28i)5-s + (−0.505 + 0.862i)7-s + (0.947 − 0.320i)9-s + (−0.987 + 0.570i)11-s + 1.53i·13-s + (0.523 − 1.38i)15-s + (0.657 + 1.13i)17-s + (−1.04 − 0.603i)19-s + (0.358 − 0.933i)21-s + (0.678 + 0.391i)23-s + (−0.601 − 1.04i)25-s + (−0.882 + 0.470i)27-s + 0.288i·29-s + (0.641 − 0.370i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.474 + 0.880i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4392207543\)
\(L(\frac12)\) \(\approx\) \(0.4392207543\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.12 - 0.843i)T \)
7 \( 1 + (9.36 - 15.9i)T \)
good5 \( 1 + (8.29 - 14.3i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (36.0 - 20.8i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 71.9iT - 2.19e3T^{2} \)
17 \( 1 + (-46.0 - 79.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (86.6 + 50.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-74.8 - 43.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 45.0iT - 2.43e4T^{2} \)
31 \( 1 + (-110. + 63.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-188. + 326. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 298.T + 6.89e4T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 + (103. - 178. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (191. - 110. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-303. - 525. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (291. + 168. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (15.4 + 26.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 450. iT - 3.57e5T^{2} \)
73 \( 1 + (-613. + 354. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-372. + 645. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + (-333. + 577. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.16e3iT - 9.12e5T^{2} \)
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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

−11.62784794918129355089652486002, −10.88954476743989260520398990715, −10.18286560276021669917639353554, −9.112498834696238004752203923229, −7.70707192103299856087154642343, −6.76340671895689473640460468020, −6.13248613136415327810403260287, −4.79292503446587780244822038100, −3.62869689402247136511878404223, −2.21476273635126428803970138278, 0.24654231081795794111351849013, 0.870505583603365147143677842766, 3.29636308509855955761436230460, 4.69812205633093419391226131833, 5.30294397856028664817302095420, 6.54821938036571040572967492117, 7.80529727138008240746806731970, 8.275758770373346572302442558871, 9.882553524151478022566699173723, 10.50637652244616400966898386693

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