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Properties

Label 2-336-84.47-c2-0-27
Degree $2$
Conductor $336$
Sign $0.0273 + 0.999i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 − 2.12i)3-s + (1.71 − 2.97i)5-s + (4.12 − 5.65i)7-s + (0.00493 − 8.99i)9-s + (3.84 + 6.65i)11-s + 5.30i·13-s + (−2.66 − 9.96i)15-s + (−11.2 − 19.4i)17-s + (−2.82 + 4.90i)19-s + (−3.22 − 20.7i)21-s + (−5.01 + 8.69i)23-s + (6.59 + 11.4i)25-s + (−19.0 − 19.1i)27-s − 44.8i·29-s + (21.3 + 37.0i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.706i)3-s + (0.343 − 0.595i)5-s + (0.589 − 0.807i)7-s + (0.000547 − 0.999i)9-s + (0.349 + 0.605i)11-s + 0.407i·13-s + (−0.177 − 0.664i)15-s + (−0.660 − 1.14i)17-s + (−0.148 + 0.257i)19-s + (−0.153 − 0.988i)21-s + (−0.218 + 0.377i)23-s + (0.263 + 0.456i)25-s + (−0.706 − 0.707i)27-s − 1.54i·29-s + (0.689 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0273 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0273 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0273 + 0.999i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.0273 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.63437 - 1.59026i\)
\(L(\frac12)\) \(\approx\) \(1.63437 - 1.59026i\)
\(L(2)\) 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 \( 1 + (-2.12 + 2.12i)T \)
7 \( 1 + (-4.12 + 5.65i)T \)
good5 \( 1 + (-1.71 + 2.97i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3.84 - 6.65i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.30iT - 169T^{2} \)
17 \( 1 + (11.2 + 19.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.82 - 4.90i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (5.01 - 8.69i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 44.8iT - 841T^{2} \)
31 \( 1 + (-21.3 - 37.0i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-19.7 + 34.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 31.1T + 1.68e3T^{2} \)
43 \( 1 - 40.4iT - 1.84e3T^{2} \)
47 \( 1 + (7.99 + 4.61i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-25.7 + 14.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (34.5 - 19.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-67.1 - 38.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-44.0 + 25.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 131.T + 5.04e3T^{2} \)
73 \( 1 + (74.3 - 42.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-77.5 - 44.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 108. iT - 6.88e3T^{2} \)
89 \( 1 + (83.6 - 144. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 14.6iT - 9.40e3T^{2} \)
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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

−11.31449366613022253585086067158, −9.915149524722830954443460759624, −9.204907211955905985002545459092, −8.237982342791044230568948656334, −7.32871475508570909042846189797, −6.52505344365403742456508360075, −4.97953079000909534827176871869, −3.93105012102763975985814192897, −2.25986594322391888691542586439, −1.06364020904109443950314867169, 2.06638576489459918843184674312, 3.14622248915556000647764432018, 4.43035404547025144252950664540, 5.60894957923961745437990520730, 6.67383246737070965436956222445, 8.227159857471025586181871250743, 8.606155244762574502584356334724, 9.715057843619996182735582464893, 10.64794644075389929912124164595, 11.25243227785138498489343520587

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