Properties

Label 2-336-7.2-c3-0-21
Degree $2$
Conductor $336$
Sign $-0.605 + 0.795i$
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 + (1.5 − 2.59i)5-s + (3.5 − 18.1i)7-s + (−4.5 + 7.79i)9-s + (−7.5 − 12.9i)11-s − 64·13-s + 9·15-s + (−42 − 72.7i)17-s + (−8 + 13.8i)19-s + (52.5 − 18.1i)21-s + (−42 + 72.7i)23-s + (58 + 100. i)25-s − 27·27-s − 297·29-s + (−126.5 − 219. i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.134 − 0.232i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.205 − 0.356i)11-s − 1.36·13-s + 0.154·15-s + (−0.599 − 1.03i)17-s + (−0.0965 + 0.167i)19-s + (0.545 − 0.188i)21-s + (−0.380 + 0.659i)23-s + (0.464 + 0.803i)25-s − 0.192·27-s − 1.90·29-s + (−0.732 − 1.26i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.605 + 0.795i$
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: \((2,\ 336,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8794411579\)
\(L(\frac12)\) \(\approx\) \(0.8794411579\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.5 + 18.1i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (7.5 + 12.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 64T + 2.19e3T^{2} \)
17 \( 1 + (42 + 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (8 - 13.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (42 - 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 297T + 2.43e4T^{2} \)
31 \( 1 + (126.5 + 219. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-158 + 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 360T + 6.89e4T^{2} \)
43 \( 1 + 26T + 7.95e4T^{2} \)
47 \( 1 + (15 - 25.9i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (181.5 + 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-59 + 102. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (185 + 320. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (181 + 313. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-233.5 + 404. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 477T + 5.71e5T^{2} \)
89 \( 1 + (453 - 784. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 503T + 9.12e5T^{2} \)
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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

−10.88440997226486244201935453340, −9.615631854361670187086026690678, −9.297560201848005976603054839226, −7.74024914171949461009663672447, −7.28160484462312649224124833904, −5.65373929054438857931024604123, −4.67858363659003933092439063930, −3.64095791833302153247079519182, −2.18006190955461941349133614785, −0.27613058830685716171005661764, 1.91406231343702784776374349737, 2.76952250924329352133802909574, 4.46211532461801271616658170915, 5.65557555992782630641101166463, 6.68118625159355183565558342623, 7.68029492314453581041215346125, 8.622555572395360024345210052951, 9.483828788112453373329368424623, 10.52245558081056606533906672721, 11.56151736026734699130438821332

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