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Properties

Label 2-336-7.3-c2-0-9
Degree $2$
Conductor $336$
Sign $0.784 + 0.620i$
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  + (−1.5 − 0.866i)3-s + (7.24 − 4.18i)5-s + (6.74 + 1.88i)7-s + (1.5 + 2.59i)9-s + (3 − 5.19i)11-s + 17.8i·13-s − 14.4·15-s + (−16.2 − 9.37i)17-s + (14.7 − 8.51i)19-s + (−8.48 − 8.66i)21-s + (6.72 + 11.6i)23-s + (22.4 − 38.9i)25-s − 5.19i·27-s + 33.9·29-s + (−12.7 − 7.37i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (1.44 − 0.836i)5-s + (0.963 + 0.268i)7-s + (0.166 + 0.288i)9-s + (0.272 − 0.472i)11-s + 1.37i·13-s − 0.965·15-s + (−0.955 − 0.551i)17-s + (0.775 − 0.447i)19-s + (−0.404 − 0.412i)21-s + (0.292 + 0.506i)23-s + (0.898 − 1.55i)25-s − 0.192i·27-s + 1.17·29-s + (−0.412 − 0.237i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.784 + 0.620i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.784 + 0.620i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.90359 - 0.661458i\)
\(L(\frac12)\) \(\approx\) \(1.90359 - 0.661458i\)
\(L(2)\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-6.74 - 1.88i)T \)
good5 \( 1 + (-7.24 + 4.18i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 17.8iT - 169T^{2} \)
17 \( 1 + (16.2 + 9.37i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.7 + 8.51i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.72 - 11.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 33.9T + 841T^{2} \)
31 \( 1 + (12.7 + 7.37i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (2.98 + 5.17i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 35.2iT - 1.68e3T^{2} \)
43 \( 1 + 15.4T + 1.84e3T^{2} \)
47 \( 1 + (-28.7 + 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-17.2 + 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (23.6 + 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (57.1 - 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 18.6T + 5.04e3T^{2} \)
73 \( 1 + (101. + 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.1 + 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 75.7iT - 6.88e3T^{2} \)
89 \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 30.5iT - 9.40e3T^{2} \)
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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

−11.46301215902271208012801830272, −10.32526293433741495828333683586, −9.103705742644743173345542214164, −8.831699087111863536847237540146, −7.25289639055717817922697493637, −6.22123382742639049479108146886, −5.29085586208094683692011100207, −4.54094877279726719753516454754, −2.24475340385583045637862320816, −1.23213891163193174025746937530, 1.49375913020863040215864666091, 2.89814070708283217346319003471, 4.58708171728656882885678640825, 5.59350061630147971114709242051, 6.42123671748674941584836250427, 7.48418616250043892695894158533, 8.740494724540405749533087323086, 9.940648739340669381952232847494, 10.47056756100046409963885200287, 11.12071823392737205599162478929

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