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Properties

Label 2-336-7.2-c3-0-0
Degree $2$
Conductor $336$
Sign $-0.662 + 0.748i$
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 + (−5.07 + 8.79i)5-s + (−4.83 + 17.8i)7-s + (−4.5 + 7.79i)9-s + (−18.9 − 32.7i)11-s − 74.0·13-s − 30.4·15-s + (22.3 + 38.6i)17-s + (69.6 − 120. i)19-s + (−53.7 + 14.2i)21-s + (28.9 − 50.0i)23-s + (10.9 + 18.9i)25-s − 27·27-s + 39.5·29-s + (−127. − 220. i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.454 + 0.786i)5-s + (−0.261 + 0.965i)7-s + (−0.166 + 0.288i)9-s + (−0.518 − 0.897i)11-s − 1.58·13-s − 0.524·15-s + (0.318 + 0.551i)17-s + (0.841 − 1.45i)19-s + (−0.558 + 0.148i)21-s + (0.262 − 0.453i)23-s + (0.0873 + 0.151i)25-s − 0.192·27-s + 0.252·29-s + (−0.738 − 1.27i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.662 + 0.748i$
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: \((2,\ 336,\ (\ :3/2),\ -0.662 + 0.748i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1484749359\)
\(L(\frac12)\) \(\approx\) \(0.1484749359\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.83 - 17.8i)T \)
good5 \( 1 + (5.07 - 8.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (18.9 + 32.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 74.0T + 2.19e3T^{2} \)
17 \( 1 + (-22.3 - 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-69.6 + 120. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-28.9 + 50.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 39.5T + 2.43e4T^{2} \)
31 \( 1 + (127. + 220. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (9.05 - 15.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 420.T + 6.89e4T^{2} \)
43 \( 1 - 382.T + 7.95e4T^{2} \)
47 \( 1 + (242. - 419. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (18.0 + 31.2i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (72.9 + 126. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (39.0 - 67.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (136. + 236. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 717.T + 3.57e5T^{2} \)
73 \( 1 + (554. + 959. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (542. - 940. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 404.T + 5.71e5T^{2} \)
89 \( 1 + (677. - 1.17e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 681.T + 9.12e5T^{2} \)
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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

−11.53726660548664832965855536687, −10.85602355584692571613287443508, −9.791003909397537428817645116512, −9.033519538041101874883875836124, −7.945635599453645209357926280241, −7.04368401553414653065277666629, −5.75286889219579869374593913586, −4.77165017387606705044821657011, −3.19599406686120794883745832941, −2.58477432346666934109672244112, 0.04963998540025867933547828281, 1.51301850842592350430045237828, 3.13747048269293780862012571525, 4.46678617279335676198333375108, 5.38225007507975364742087632458, 7.19614733141983968163309855501, 7.40500049708093498257565705454, 8.510518890275054217395541964950, 9.762724283105315477405447903191, 10.24939239587828307671489164424

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