Properties

Label 2-3822-13.12-c1-0-89
Degree $2$
Conductor $3822$
Sign $-0.832 + 0.554i$
Analytic cond. $30.5188$
Root an. cond. $5.52438$
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  + i·2-s − 3-s − 4-s − 2i·5-s i·6-s i·8-s + 9-s + 2·10-s + 12-s + (−2 − 3i)13-s + 2i·15-s + 16-s + 2·17-s + i·18-s − 4i·19-s + 2i·20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.894i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s + 0.632·10-s + 0.288·12-s + (−0.554 − 0.832i)13-s + 0.516i·15-s + 0.250·16-s + 0.485·17-s + 0.235i·18-s − 0.917i·19-s + 0.447i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3822\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(30.5188\)
Root analytic conductor: \(5.52438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3822} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3822,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3913804180\)
\(L(\frac12)\) \(\approx\) \(0.3913804180\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + T \)
7 \( 1 \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 4iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.122355185600567942613201423498, −7.45266808206281926041915776106, −6.73034273168710720939630488159, −5.78026219505288544744416246735, −5.33462430195331654459716392519, −4.62996282583798631118008252949, −3.84849602046665846154564978220, −2.57569096812133945889563895155, −1.14569926967265021163286808671, −0.13835932908597071675205284041, 1.43978161865337530302982254744, 2.36445560711683178088318063611, 3.31645978133578997846495246499, 4.13427384757802814094378207669, 4.89960585548359665993483001124, 5.92106320490358142633258051317, 6.42387833340555880189861123424, 7.39015759565747129628504855459, 7.930285637752151492286165146755, 8.998310390228690436984839026373

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