Properties

Label 2-3822-13.12-c1-0-53
Degree $2$
Conductor $3822$
Sign $0.554 + 0.832i$
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 + (3 − 2i)13-s − 2i·15-s + 16-s + 2·17-s i·18-s − 6i·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 + (0.832 − 0.554i)13-s − 0.516i·15-s + 0.250·16-s + 0.485·17-s − 0.235i·18-s − 1.37i·19-s − 0.447i·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3822\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $0.554 + 0.832i$
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: \((2,\ 3822,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.408352940\)
\(L(\frac12)\) \(\approx\) \(1.408352940\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 2i)T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 10iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−8.592443095006908719630686474791, −7.43831754561200452866466859406, −6.99727455278791735341603897689, −6.06845525461585603333445897033, −5.34711438141615528898344326510, −4.57151468145937373871369189002, −3.43753072507114542699678702954, −3.01230306631566244689649768439, −1.79593973989094017552982170993, −0.63532087720513272905995758472, 0.844615223775748613531646780089, 1.79502897079381486437847821399, 3.48681924873775414111823132062, 4.21043966905202273620386678418, 4.97300766978274485378060510492, 5.81251956109654462293586506546, 6.11725242134247229214825955898, 7.18606805449285450629387747142, 7.83144886322286838288380768687, 8.508678442950309649005647619017

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