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Properties

Label 2-336-21.17-c3-0-21
Degree $2$
Conductor $336$
Sign $0.889 - 0.456i$
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  + (−3.62 + 3.72i)3-s + (0.263 + 0.456i)5-s + (16.6 + 8.05i)7-s + (−0.685 − 26.9i)9-s + (9.84 + 5.68i)11-s − 65.8i·13-s + (−2.65 − 0.674i)15-s + (0.339 − 0.587i)17-s + (19.0 − 10.9i)19-s + (−90.4 + 32.8i)21-s + (6.60 − 3.81i)23-s + (62.3 − 108. i)25-s + (102. + 95.3i)27-s + 165. i·29-s + (110. + 63.6i)31-s + ⋯
L(s)  = 1  + (−0.698 + 0.716i)3-s + (0.0235 + 0.0407i)5-s + (0.900 + 0.435i)7-s + (−0.0253 − 0.999i)9-s + (0.269 + 0.155i)11-s − 1.40i·13-s + (−0.0456 − 0.0116i)15-s + (0.00483 − 0.00837i)17-s + (0.229 − 0.132i)19-s + (−0.940 + 0.340i)21-s + (0.0598 − 0.0345i)23-s + (0.498 − 0.864i)25-s + (0.733 + 0.679i)27-s + 1.06i·29-s + (0.639 + 0.369i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.889 - 0.456i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.889 - 0.456i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.642159001\)
\(L(\frac12)\) \(\approx\) \(1.642159001\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.62 - 3.72i)T \)
7 \( 1 + (-16.6 - 8.05i)T \)
good5 \( 1 + (-0.263 - 0.456i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-9.84 - 5.68i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 65.8iT - 2.19e3T^{2} \)
17 \( 1 + (-0.339 + 0.587i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-19.0 + 10.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.60 + 3.81i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 165. iT - 2.43e4T^{2} \)
31 \( 1 + (-110. - 63.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (95.9 + 166. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 508.T + 6.89e4T^{2} \)
43 \( 1 - 25.2T + 7.95e4T^{2} \)
47 \( 1 + (-167. - 289. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-376. - 217. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (208. - 361. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (288. - 166. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-68.6 + 118. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 718. iT - 3.57e5T^{2} \)
73 \( 1 + (-1.00e3 - 582. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (189. + 327. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 539.T + 5.71e5T^{2} \)
89 \( 1 + (-457. - 792. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 29.2iT - 9.12e5T^{2} \)
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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

−10.93062340549559556074318246200, −10.57728470070032019974740352885, −9.388058949862083049438184851043, −8.505159095006370457817369794528, −7.38061506764795848458285992353, −6.03903270897564138116624941930, −5.24242619317812302102158473472, −4.30174281474544583651894906239, −2.85231436302586988029271690924, −0.921886128898117008419042879795, 0.969550684893816956663191660216, 2.11831327944877487892425206733, 4.12539194086943405028136577118, 5.11627596010266784780715784645, 6.28360609609356344925770082210, 7.17605986474935913633904160954, 8.023809188329207019695701222025, 9.126713265804667393752274107747, 10.34239407844741535887453975061, 11.43110922682190536512942146400

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