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Properties

Label 2-392-56.51-c2-0-36
Degree $2$
Conductor $392$
Sign $0.942 - 0.335i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
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  + (−0.288 + 1.97i)2-s + (0.0487 + 0.0843i)3-s + (−3.83 − 1.14i)4-s + (3.00 + 1.73i)5-s + (−0.181 + 0.0720i)6-s + (3.37 − 7.25i)8-s + (4.49 − 7.78i)9-s + (−4.29 + 5.43i)10-s + (1.46 + 2.53i)11-s + (−0.0902 − 0.379i)12-s − 19.1i·13-s + 0.337i·15-s + (13.3 + 8.76i)16-s + (−7.19 − 12.4i)17-s + (14.1 + 11.1i)18-s + (4.04 − 7.01i)19-s + ⋯
L(s)  = 1  + (−0.144 + 0.989i)2-s + (0.0162 + 0.0281i)3-s + (−0.958 − 0.285i)4-s + (0.600 + 0.346i)5-s + (−0.0301 + 0.0120i)6-s + (0.421 − 0.906i)8-s + (0.499 − 0.865i)9-s + (−0.429 + 0.543i)10-s + (0.133 + 0.230i)11-s + (−0.00751 − 0.0315i)12-s − 1.47i·13-s + 0.0225i·15-s + (0.836 + 0.548i)16-s + (−0.423 − 0.733i)17-s + (0.783 + 0.619i)18-s + (0.213 − 0.369i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.942 - 0.335i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.56008 + 0.269471i\)
\(L(\frac12)\) \(\approx\) \(1.56008 + 0.269471i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.288 - 1.97i)T \)
7 \( 1 \)
good3 \( 1 + (-0.0487 - 0.0843i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.00 - 1.73i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.46 - 2.53i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 19.1iT - 169T^{2} \)
17 \( 1 + (7.19 + 12.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.04 + 7.01i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.5 - 8.37i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 27.1iT - 841T^{2} \)
31 \( 1 + (-38.8 + 22.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-34.2 - 19.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 45.8T + 1.68e3T^{2} \)
43 \( 1 - 61.0T + 1.84e3T^{2} \)
47 \( 1 + (-40.0 - 23.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (8.39 - 4.84i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (57.2 + 99.2i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.47 - 3.74i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.02 - 10.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 129. iT - 5.04e3T^{2} \)
73 \( 1 + (9.14 + 15.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (36.9 + 21.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 109.T + 6.88e3T^{2} \)
89 \( 1 + (40.4 - 70.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 162.T + 9.40e3T^{2} \)
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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

−10.87972280800789631998997962702, −9.876610215062645105850560849074, −9.404450135910218173938919469476, −8.265546041841538231478322658618, −7.25034249251338946532342317638, −6.46371643049200809969273005372, −5.54736366760467517216195223624, −4.44440415954425405912843965439, −3.00341368152767411947253165884, −0.844874282571956507355586132846, 1.40318957077855367834269812886, 2.38371514487879422841599697369, 4.04113069314209830798628397603, 4.85072851806235855573645774137, 6.12563508951634601248704531331, 7.50011881047499276765094838353, 8.615791521402545999026332632435, 9.322681257622294529335922288536, 10.20371001541842958052500731295, 10.98527787659833090948728723727

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