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Properties

Label 2-336-7.3-c2-0-11
Degree $2$
Conductor $336$
Sign $-0.319 + 0.947i$
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 + (−8.29 + 4.79i)5-s + (−0.5 − 6.98i)7-s + (1.5 + 2.59i)9-s + (2.29 − 3.97i)11-s − 7.84i·13-s − 16.5·15-s + (−16.5 − 9.58i)17-s + (14.2 − 8.20i)19-s + (5.29 − 10.9i)21-s + (−12 − 20.7i)23-s + (33.3 − 57.8i)25-s + 5.19i·27-s − 10.5·29-s + (−48.2 − 27.8i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (−1.65 + 0.958i)5-s + (−0.0714 − 0.997i)7-s + (0.166 + 0.288i)9-s + (0.208 − 0.361i)11-s − 0.603i·13-s − 1.10·15-s + (−0.976 − 0.563i)17-s + (0.747 − 0.431i)19-s + (0.252 − 0.519i)21-s + (−0.521 − 0.903i)23-s + (1.33 − 2.31i)25-s + 0.192i·27-s − 0.365·29-s + (−1.55 − 0.899i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.319 + 0.947i$
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: \((2,\ 336,\ (\ :1),\ -0.319 + 0.947i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.386525 - 0.538320i\)
\(L(\frac12)\) \(\approx\) \(0.386525 - 0.538320i\)
\(L(2)\) 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 \( 1 + (0.5 + 6.98i)T \)
good5 \( 1 + (8.29 - 4.79i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.29 + 3.97i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 7.84iT - 169T^{2} \)
17 \( 1 + (16.5 + 9.58i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.2 + 8.20i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (12 + 20.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 10.5T + 841T^{2} \)
31 \( 1 + (48.2 + 27.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-12.2 - 21.1i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 48.7iT - 1.68e3T^{2} \)
43 \( 1 - 18.7T + 1.84e3T^{2} \)
47 \( 1 + (-10.4 + 6.00i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-18.7 + 32.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (54.7 + 31.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (96.3 - 55.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-31.9 + 55.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 21.5T + 5.04e3T^{2} \)
73 \( 1 + (53.5 + 30.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (1.90 + 3.30i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 100. iT - 6.88e3T^{2} \)
89 \( 1 + (-85.7 + 49.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 63.5iT - 9.40e3T^{2} \)
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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

−11.05313572783309405677200187132, −10.36837615155372663532302815059, −9.140738913207466427635201758789, −7.936785024557267722615977680450, −7.46393269432068006642805107332, −6.51034439032143094208341305086, −4.59378971911858087046125180163, −3.76306467633733172666879584856, −2.88126746569171385918090468967, −0.28973834727422522716543048228, 1.72503431728619745228009509142, 3.49195086411855691442841683102, 4.38300279889728084697984067712, 5.62876049485506154671223033074, 7.16365741081081569635676104490, 7.86812775997552542294110611541, 8.933161258957606423587995718381, 9.186474542874880835729303714726, 10.97197294089830371077135126983, 11.90427738701699699623028358508

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