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Properties

Label 2-336-7.3-c6-0-15
Degree $2$
Conductor $336$
Sign $0.927 - 0.373i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.5 − 7.79i)3-s + (−167. + 96.4i)5-s + (243. − 241. i)7-s + (121.5 + 210. i)9-s + (−853. + 1.47e3i)11-s − 3.48e3i·13-s + 3.00e3·15-s + (−1.01e3 − 587. i)17-s + (−5.70e3 + 3.29e3i)19-s + (−5.17e3 + 1.35e3i)21-s + (−1.41e3 − 2.44e3i)23-s + (1.07e4 − 1.86e4i)25-s − 3.78e3i·27-s − 3.80e4·29-s + (−7.41e3 − 4.27e3i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−1.33 + 0.771i)5-s + (0.710 − 0.703i)7-s + (0.166 + 0.288i)9-s + (−0.641 + 1.11i)11-s − 1.58i·13-s + 0.890·15-s + (−0.207 − 0.119i)17-s + (−0.831 + 0.480i)19-s + (−0.558 + 0.146i)21-s + (−0.116 − 0.201i)23-s + (0.690 − 1.19i)25-s − 0.192i·27-s − 1.56·29-s + (−0.248 − 0.143i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ 0.927 - 0.373i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.7384879272\)
\(L(\frac12)\) \(\approx\) \(0.7384879272\)
\(L(4)\) 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 \( 1 + (13.5 + 7.79i)T \)
7 \( 1 + (-243. + 241. i)T \)
good5 \( 1 + (167. - 96.4i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (853. - 1.47e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 3.48e3iT - 4.82e6T^{2} \)
17 \( 1 + (1.01e3 + 587. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (5.70e3 - 3.29e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (1.41e3 + 2.44e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 3.80e4T + 5.94e8T^{2} \)
31 \( 1 + (7.41e3 + 4.27e3i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (3.89e4 + 6.74e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 8.67e3iT - 4.75e9T^{2} \)
43 \( 1 - 6.76e4T + 6.32e9T^{2} \)
47 \( 1 + (6.50e4 - 3.75e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (5.22e4 - 9.04e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.09e5 - 6.34e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (7.51e3 - 4.33e3i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (2.01e5 - 3.49e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 5.85e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.80e5 - 2.19e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.74e5 - 4.75e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 8.96e5iT - 3.26e11T^{2} \)
89 \( 1 + (7.10e5 - 4.10e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.21e6iT - 8.32e11T^{2} \)
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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

−10.83194559997680263065108825609, −10.05542038455359484820495640270, −8.290582275862805913457800673756, −7.52950849519830846924460068522, −7.17564919401187147827350271367, −5.66931345149215834330970833823, −4.52863863829319699154130490743, −3.59689235260945086938509851231, −2.14844314979307317915116876627, −0.52088961541047117603371703026, 0.36793496784711099151326711624, 1.86473449694360329688632801927, 3.60326499913488055069706767975, 4.55661049379812394445041326510, 5.32424921446962889224921958209, 6.57564076729698563239827361117, 7.85522146191422672986176674845, 8.597631123002305306013899960906, 9.278718332537583188701392238366, 10.94237120583213268542096013668

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