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Properties

Label 2-828-92.91-c1-0-27
Degree $2$
Conductor $828$
Sign $-0.625 - 0.780i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
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  + (1 + i)2-s + 2i·4-s + 3.74i·5-s + 3.74·7-s + (−2 + 2i)8-s + (−3.74 + 3.74i)10-s + 3.74·11-s − 13-s + (3.74 + 3.74i)14-s − 4·16-s − 3.74i·17-s − 7.48·20-s + (3.74 + 3.74i)22-s + (3.74 − 3i)23-s − 9·25-s + (−1 − i)26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + 1.67i·5-s + 1.41·7-s + (−0.707 + 0.707i)8-s + (−1.18 + 1.18i)10-s + 1.12·11-s − 0.277·13-s + (0.999 + 0.999i)14-s − 16-s − 0.907i·17-s − 1.67·20-s + (0.797 + 0.797i)22-s + (0.780 − 0.625i)23-s − 1.80·25-s + (−0.196 − 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.625 - 0.780i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ -0.625 - 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12739 + 2.34896i\)
\(L(\frac12)\) \(\approx\) \(1.12739 + 2.34896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
23 \( 1 + (-3.74 + 3i)T \)
good5 \( 1 - 3.74iT - 5T^{2} \)
7 \( 1 - 3.74T + 7T^{2} \)
11 \( 1 - 3.74T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 3.74iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 - 3.74iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 7.48T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 7.48iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 3.74T + 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 - 7.48iT - 89T^{2} \)
97 \( 1 - 3.74iT - 97T^{2} \)
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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

−10.81936247739090859244287320621, −9.631332715640950942435051130931, −8.577883564264478834018696893892, −7.61825150431914491782487722424, −7.03703081317055081371438995021, −6.32788747375092256402138732341, −5.23028204796793328978268883767, −4.27115780555630128265254159581, −3.24413742629649869235049228443, −2.15995380320348188779435304719, 1.19787875660570498788230934773, 1.80200334843326956814561574800, 3.71165633207846380505582890524, 4.56345159694484047218865599742, 5.13510009459456948142226344017, 5.99715240592124126902050623347, 7.39912362746177382838090066232, 8.568872899139357909736062675203, 9.015332042561939707268788692001, 9.930703567495323716519575841830

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