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Properties

Label 2-336-21.5-c3-0-14
Degree $2$
Conductor $336$
Sign $-0.562 - 0.826i$
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  + (1.40 + 5.00i)3-s + (−0.263 + 0.456i)5-s + (16.6 − 8.05i)7-s + (−23.0 + 14.0i)9-s + (−9.84 + 5.68i)11-s + 65.8i·13-s + (−2.65 − 0.674i)15-s + (−0.339 − 0.587i)17-s + (19.0 + 10.9i)19-s + (63.7 + 72.0i)21-s + (−6.60 − 3.81i)23-s + (62.3 + 108. i)25-s + (−102. − 95.3i)27-s + 165. i·29-s + (110. − 63.6i)31-s + ⋯
L(s)  = 1  + (0.271 + 0.962i)3-s + (−0.0235 + 0.0407i)5-s + (0.900 − 0.435i)7-s + (−0.853 + 0.521i)9-s + (−0.269 + 0.155i)11-s + 1.40i·13-s + (−0.0456 − 0.0116i)15-s + (−0.00483 − 0.00837i)17-s + (0.229 + 0.132i)19-s + (0.662 + 0.748i)21-s + (−0.0598 − 0.0345i)23-s + (0.498 + 0.864i)25-s + (−0.733 − 0.679i)27-s + 1.06i·29-s + (0.639 − 0.369i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.562 - 0.826i$
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: \((2,\ 336,\ (\ :3/2),\ -0.562 - 0.826i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.784969375\)
\(L(\frac12)\) \(\approx\) \(1.784969375\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.40 - 5.00i)T \)
7 \( 1 + (-16.6 + 8.05i)T \)
good5 \( 1 + (0.263 - 0.456i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (9.84 - 5.68i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 65.8iT - 2.19e3T^{2} \)
17 \( 1 + (0.339 + 0.587i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.0 - 10.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (6.60 + 3.81i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 165. iT - 2.43e4T^{2} \)
31 \( 1 + (-110. + 63.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (95.9 - 166. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 508.T + 6.89e4T^{2} \)
43 \( 1 - 25.2T + 7.95e4T^{2} \)
47 \( 1 + (167. - 289. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (376. - 217. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-208. - 361. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (288. + 166. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-68.6 - 118. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 718. iT - 3.57e5T^{2} \)
73 \( 1 + (-1.00e3 + 582. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (189. - 327. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 539.T + 5.71e5T^{2} \)
89 \( 1 + (457. - 792. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 29.2iT - 9.12e5T^{2} \)
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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

−11.26265091253787854128636827107, −10.56792096892130947519090062222, −9.582878964304842756288931641868, −8.745106187304313620341260795744, −7.81803476290497440530215652124, −6.67523689973756139058452745882, −5.14814893354431542075710860517, −4.48902356393471112776860165845, −3.30166962897030959424450820638, −1.71754531000241599102961250321, 0.61363692747245563867734030454, 2.06978768023401074081247995672, 3.21731574348510141927777274934, 4.98580240902901041574278553945, 5.90054975538111108986126249671, 7.07084768898265255406206109074, 8.232178737911611996228371784289, 8.388477615524199799843951733610, 9.887860123038682064926460167061, 10.96733383533579576225755072982

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