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Properties

Label 2-392-392.221-c1-0-24
Degree $2$
Conductor $392$
Sign $-0.0859 + 0.996i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−0.364 − 1.36i)2-s + (−2.07 + 0.816i)3-s + (−1.73 + 0.996i)4-s + (0.313 − 2.08i)5-s + (1.87 + 2.54i)6-s + (1.56 + 2.13i)7-s + (1.99 + 2.00i)8-s + (1.45 − 1.35i)9-s + (−2.95 + 0.330i)10-s + (−0.0395 + 0.0426i)11-s + (2.79 − 3.48i)12-s + (0.166 + 0.0379i)13-s + (2.33 − 2.91i)14-s + (1.04 + 4.58i)15-s + (2.01 − 3.45i)16-s + (0.444 − 5.93i)17-s + ⋯
L(s)  = 1  + (−0.257 − 0.966i)2-s + (−1.20 + 0.471i)3-s + (−0.867 + 0.498i)4-s + (0.140 − 0.931i)5-s + (0.764 + 1.03i)6-s + (0.592 + 0.805i)7-s + (0.704 + 0.709i)8-s + (0.486 − 0.451i)9-s + (−0.935 + 0.104i)10-s + (−0.0119 + 0.0128i)11-s + (0.806 − 1.00i)12-s + (0.0461 + 0.0105i)13-s + (0.625 − 0.780i)14-s + (0.270 + 1.18i)15-s + (0.503 − 0.863i)16-s + (0.107 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0859 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0859 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0859 + 0.996i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.0859 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492993 - 0.537330i\)
\(L(\frac12)\) \(\approx\) \(0.492993 - 0.537330i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.364 + 1.36i)T \)
7 \( 1 + (-1.56 - 2.13i)T \)
good3 \( 1 + (2.07 - 0.816i)T + (2.19 - 2.04i)T^{2} \)
5 \( 1 + (-0.313 + 2.08i)T + (-4.77 - 1.47i)T^{2} \)
11 \( 1 + (0.0395 - 0.0426i)T + (-0.822 - 10.9i)T^{2} \)
13 \( 1 + (-0.166 - 0.0379i)T + (11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.444 + 5.93i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-2.28 + 1.31i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.126 + 1.69i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (2.96 + 6.15i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 + (-0.742 + 1.28i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.404 - 0.592i)T + (-13.5 - 34.4i)T^{2} \)
41 \( 1 + (-6.94 + 8.70i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (2.90 - 2.31i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-8.42 + 2.59i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (-4.38 - 6.43i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-0.891 - 5.91i)T + (-56.3 + 17.3i)T^{2} \)
61 \( 1 + (-4.44 + 6.51i)T + (-22.2 - 56.7i)T^{2} \)
67 \( 1 + (-12.9 - 7.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.10 + 1.97i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (7.59 + 2.34i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (7.38 + 12.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.21 + 0.277i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (-2.03 + 1.88i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−11.29976785888092063413979164309, −10.29213890566600388012857953955, −9.343825448563235696314469671888, −8.729832483337460364338974832431, −7.52488858065617844944641341044, −5.71155213111014357020219037401, −5.09955747166728750904347561005, −4.31952795900731190617507018745, −2.47698796868303863266002321397, −0.73855497303946083699116092451, 1.26948474758579655199773355153, 3.79913863729754477872366199770, 5.14469840170478532646033972424, 6.01224974455899005782494056676, 6.81994568934548648769306624250, 7.46827227331558465553915842961, 8.493531876321356643038432099037, 9.930218943042488691588154737451, 10.70602989661522585575987591384, 11.22950256182193187681601328892

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