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Properties

Label 2-336-7.2-c3-0-1
Degree $2$
Conductor $336$
Sign $-0.701 - 0.712i$
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  + (−1.5 − 2.59i)3-s + (−3.5 + 6.06i)5-s + (17.5 + 6.06i)7-s + (−4.5 + 7.79i)9-s + (3.5 + 6.06i)11-s − 52·13-s + 21·15-s + (−36 − 62.3i)17-s + (10 − 17.3i)19-s + (−10.5 − 54.5i)21-s + (−24 + 41.5i)23-s + (38 + 65.8i)25-s + 27·27-s − 243·29-s + (47.5 + 82.2i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.313 + 0.542i)5-s + (0.944 + 0.327i)7-s + (−0.166 + 0.288i)9-s + (0.0959 + 0.166i)11-s − 1.10·13-s + 0.361·15-s + (−0.513 − 0.889i)17-s + (0.120 − 0.209i)19-s + (−0.109 − 0.566i)21-s + (−0.217 + 0.376i)23-s + (0.303 + 0.526i)25-s + 0.192·27-s − 1.55·29-s + (0.275 + 0.476i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5923429591\)
\(L(\frac12)\) \(\approx\) \(0.5923429591\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 + (-17.5 - 6.06i)T \)
good5 \( 1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-3.5 - 6.06i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 52T + 2.19e3T^{2} \)
17 \( 1 + (36 + 62.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10 + 17.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (24 - 41.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 243T + 2.43e4T^{2} \)
31 \( 1 + (-47.5 - 82.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (176 - 304. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 296T + 6.89e4T^{2} \)
43 \( 1 + 158T + 7.95e4T^{2} \)
47 \( 1 + (71 - 122. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-187.5 - 324. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-139.5 - 241. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (123 - 213. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (365 + 632. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 338T + 3.57e5T^{2} \)
73 \( 1 + (-271 - 469. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (152.5 - 264. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.12e3T + 5.71e5T^{2} \)
89 \( 1 + (-213 + 368. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 369T + 9.12e5T^{2} \)
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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

−11.61539604215763249739254180093, −10.77236707031257093073762291445, −9.635399519932249496501648038808, −8.542942690729029615989297886583, −7.44032182163567959265361623952, −6.95088234118917892805002856297, −5.48923003553805115382807850939, −4.64102308332650932047665131636, −2.95793528559440532826766385016, −1.68756826127567366277098844379, 0.21074069435558627893870441867, 1.94189992670889740080811254251, 3.82742139000405564426332305807, 4.69685100634529141342710886258, 5.58562915736206628470840563453, 6.99126396212536506782715607536, 8.061353232996184694510439069890, 8.836370075062313285927782482062, 9.971478098983298193625478616046, 10.79390716009387905853506156502

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