Properties

Label 2-336-84.23-c1-0-15
Degree $2$
Conductor $336$
Sign $-0.994 + 0.103i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.396 − 1.68i)3-s + (−2.87 − 1.65i)5-s + (1.73 − 2i)7-s + (−2.68 + 1.33i)9-s + (1.65 + 2.87i)11-s − 4·13-s + (−1.65 + 5.5i)15-s + (−2.87 + 1.65i)17-s + (−6.06 − 3.5i)19-s + (−4.05 − 2.12i)21-s + (1.65 − 2.87i)23-s + (3 + 5.19i)25-s + (3.31 + 4i)27-s − 6.63i·29-s + (2.59 − 1.5i)31-s + ⋯
L(s)  = 1  + (−0.228 − 0.973i)3-s + (−1.28 − 0.741i)5-s + (0.654 − 0.755i)7-s + (−0.895 + 0.445i)9-s + (0.500 + 0.866i)11-s − 1.10·13-s + (−0.428 + 1.42i)15-s + (−0.696 + 0.402i)17-s + (−1.39 − 0.802i)19-s + (−0.885 − 0.464i)21-s + (0.345 − 0.598i)23-s + (0.600 + 1.03i)25-s + (0.638 + 0.769i)27-s − 1.23i·29-s + (0.466 − 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.994 + 0.103i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0312766 - 0.600899i\)
\(L(\frac12)\) \(\approx\) \(0.0312766 - 0.600899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.396 + 1.68i)T \)
7 \( 1 + (-1.73 + 2i)T \)
good5 \( 1 + (2.87 + 1.65i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.65 - 2.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (2.87 - 1.65i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.06 + 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.65 + 2.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.63iT - 29T^{2} \)
31 \( 1 + (-2.59 + 1.5i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.63iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-4.97 + 8.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.87 + 1.65i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.65 + 2.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.79 + 4.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 + 4.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-2.87 - 1.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−11.43413926891495740384643674997, −10.41574650135665196186826867228, −8.905205983894288948970315659764, −8.088139938203230809635797129797, −7.36082152523876628815002773427, −6.56225801601871687590142179222, −4.77634433334783461924630185088, −4.26135540380764316530326904408, −2.15055267506927584638255289964, −0.42318459398530991891274617564, 2.77919378883500755212716798197, 3.91587627255252472078617711815, 4.86863164848196438843490050539, 6.08529495988584377167802529159, 7.30086834492628643414445594411, 8.426213359101306026884296825579, 9.077642338030976807253640307362, 10.43111886322429683613601033157, 11.12831562533996367072049985197, 11.72354934338733543921311388304

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