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Properties

Label 2-336-84.11-c3-0-24
Degree $2$
Conductor $336$
Sign $-0.0402 + 0.999i$
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  + (−2.73 − 4.41i)3-s + (−17.5 + 10.1i)5-s + (18.5 + 0.135i)7-s + (−12.0 + 24.1i)9-s + (−1.55 + 2.69i)11-s + 32.4·13-s + (92.7 + 49.8i)15-s + (−14.0 − 8.09i)17-s + (−66.3 + 38.3i)19-s + (−50.0 − 82.2i)21-s + (−16.6 − 28.8i)23-s + (142. − 247. i)25-s + (139. − 12.7i)27-s − 173. i·29-s + (164. + 94.7i)31-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)3-s + (−1.56 + 0.906i)5-s + (0.999 + 0.00733i)7-s + (−0.446 + 0.894i)9-s + (−0.0426 + 0.0738i)11-s + 0.691·13-s + (1.59 + 0.858i)15-s + (−0.200 − 0.115i)17-s + (−0.801 + 0.462i)19-s + (−0.519 − 0.854i)21-s + (−0.151 − 0.261i)23-s + (1.14 − 1.97i)25-s + (0.995 − 0.0911i)27-s − 1.11i·29-s + (0.950 + 0.548i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0402 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0402 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.0402 + 0.999i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.0402 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8270235581\)
\(L(\frac12)\) \(\approx\) \(0.8270235581\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.73 + 4.41i)T \)
7 \( 1 + (-18.5 - 0.135i)T \)
good5 \( 1 + (17.5 - 10.1i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (1.55 - 2.69i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 32.4T + 2.19e3T^{2} \)
17 \( 1 + (14.0 + 8.09i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (66.3 - 38.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (16.6 + 28.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 173. iT - 2.43e4T^{2} \)
31 \( 1 + (-164. - 94.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (129. + 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 149. iT - 6.89e4T^{2} \)
43 \( 1 + 495. iT - 7.95e4T^{2} \)
47 \( 1 + (236. + 409. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (482. + 278. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (262. - 454. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (111. + 192. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-241. - 139. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 + (-144. + 250. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-641. + 370. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 628.T + 5.71e5T^{2} \)
89 \( 1 + (-1.13e3 + 657. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.22e3T + 9.12e5T^{2} \)
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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

−11.02074264995956544549887179053, −10.51740982372915204564587587640, −8.445933321791895064632611613278, −8.011535742966321548244390828318, −7.10400295938350422996240312931, −6.25748604689855092003550039965, −4.81135989088951568428165486615, −3.68910220919686416613456655125, −2.12520824456290103445081869533, −0.40181938338946792387636419800, 1.03708501733820435022510019122, 3.46818942232234371569132400960, 4.50261445499255186829236927427, 4.96037492773751440087473932103, 6.42909355925201276650507337856, 7.915827534910358147338495401061, 8.462621619994286814444527801945, 9.370973371855183264049822244935, 10.91175913740302562019781332472, 11.18033344976167951524125994158

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