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Properties

Label 2-336-112.61-c2-0-27
Degree $2$
Conductor $336$
Sign $0.997 + 0.0742i$
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  + (−1.94 − 0.455i)2-s + (1.67 + 0.448i)3-s + (3.58 + 1.77i)4-s + (−6.86 + 1.83i)5-s + (−3.05 − 1.63i)6-s + (6.65 − 2.17i)7-s + (−6.17 − 5.08i)8-s + (2.59 + 1.50i)9-s + (14.2 − 0.457i)10-s + (−9.67 − 2.59i)11-s + (5.20 + 4.57i)12-s + (8.11 − 8.11i)13-s + (−13.9 + 1.21i)14-s − 12.3·15-s + (9.71 + 12.7i)16-s + (17.0 − 9.86i)17-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)2-s + (0.557 + 0.149i)3-s + (0.896 + 0.443i)4-s + (−1.37 + 0.367i)5-s + (−0.509 − 0.272i)6-s + (0.950 − 0.311i)7-s + (−0.771 − 0.635i)8-s + (0.288 + 0.166i)9-s + (1.42 − 0.0457i)10-s + (−0.879 − 0.235i)11-s + (0.433 + 0.381i)12-s + (0.624 − 0.624i)13-s + (−0.996 + 0.0866i)14-s − 0.820·15-s + (0.607 + 0.794i)16-s + (1.00 − 0.580i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.997 + 0.0742i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.997 + 0.0742i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.13714 - 0.0422679i\)
\(L(\frac12)\) \(\approx\) \(1.13714 - 0.0422679i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.94 + 0.455i)T \)
3 \( 1 + (-1.67 - 0.448i)T \)
7 \( 1 + (-6.65 + 2.17i)T \)
good5 \( 1 + (6.86 - 1.83i)T + (21.6 - 12.5i)T^{2} \)
11 \( 1 + (9.67 + 2.59i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (-8.11 + 8.11i)T - 169iT^{2} \)
17 \( 1 + (-17.0 + 9.86i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.49 - 5.58i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-38.8 - 22.4i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-10.8 - 10.8i)T + 841iT^{2} \)
31 \( 1 + (20.6 - 11.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.325 + 1.21i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 - 41.1T + 1.68e3T^{2} \)
43 \( 1 + (-15.3 + 15.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-73.8 - 42.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (2.36 + 0.634i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-26.1 + 97.6i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-17.1 - 64.1i)T + (-3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-22.5 + 84.1i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 100. iT - 5.04e3T^{2} \)
73 \( 1 + (-19.3 - 33.5i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (16.0 - 27.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (28.6 - 28.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (11.3 - 19.6i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 134. iT - 9.40e3T^{2} \)
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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

−10.89234062738061589513336160209, −10.76591796963504164873066533548, −9.342211881333486966176959048982, −8.317858443427552075235923510984, −7.70962858585662520352996609085, −7.22926616188474342463285347040, −5.32486792736788164068716825260, −3.73764707590985998533726591038, −2.90462065072275621897362201595, −0.983213013032779852675886934756, 1.00952064294580154701875563259, 2.61607071476220261794648542693, 4.18180970593714273957906791771, 5.47564460675221999069412421137, 7.06980612899639443030366364892, 7.79608418981554320617845763695, 8.442912808088933127862530438103, 9.076090816241115068892381600992, 10.48638272279259605791415177847, 11.22971249079634858394127300414

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