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Properties

Label 2-828-12.11-c1-0-23
Degree $2$
Conductor $828$
Sign $0.614 - 0.789i$
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  + (1.13 + 0.842i)2-s + (0.579 + 1.91i)4-s − 1.27i·5-s − 0.887i·7-s + (−0.955 + 2.66i)8-s + (1.07 − 1.44i)10-s + 2.82·11-s + 5.21·13-s + (0.748 − 1.00i)14-s + (−3.32 + 2.21i)16-s + 2.72i·17-s − 0.887i·19-s + (2.43 − 0.736i)20-s + (3.21 + 2.38i)22-s − 23-s + ⋯
L(s)  = 1  + (0.803 + 0.595i)2-s + (0.289 + 0.957i)4-s − 0.568i·5-s − 0.335i·7-s + (−0.337 + 0.941i)8-s + (0.338 − 0.456i)10-s + 0.852·11-s + 1.44·13-s + (0.199 − 0.269i)14-s + (−0.832 + 0.554i)16-s + 0.661i·17-s − 0.203i·19-s + (0.544 − 0.164i)20-s + (0.684 + 0.508i)22-s − 0.208·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.614 - 0.789i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36932 + 1.15832i\)
\(L(\frac12)\) \(\approx\) \(2.36932 + 1.15832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.842i)T \)
3 \( 1 \)
23 \( 1 + T \)
good5 \( 1 + 1.27iT - 5T^{2} \)
7 \( 1 + 0.887iT - 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 - 5.21T + 13T^{2} \)
17 \( 1 - 2.72iT - 17T^{2} \)
19 \( 1 + 0.887iT - 19T^{2} \)
29 \( 1 - 0.585iT - 29T^{2} \)
31 \( 1 - 4.22iT - 31T^{2} \)
37 \( 1 - 5.57T + 37T^{2} \)
41 \( 1 + 2.40iT - 41T^{2} \)
43 \( 1 - 6.36iT - 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + 9.06iT - 53T^{2} \)
59 \( 1 + 7.41T + 59T^{2} \)
61 \( 1 + 0.851T + 61T^{2} \)
67 \( 1 + 8.19iT - 67T^{2} \)
71 \( 1 + 7.91T + 71T^{2} \)
73 \( 1 + 7.25T + 73T^{2} \)
79 \( 1 + 14.5iT - 79T^{2} \)
83 \( 1 + 1.45T + 83T^{2} \)
89 \( 1 - 3.00iT - 89T^{2} \)
97 \( 1 - 9.45T + 97T^{2} \)
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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

−10.53177378839602568766442638382, −9.154058887140851072155248055127, −8.568769954428089092813680977902, −7.74745677671779566604483095177, −6.59667520640096422285781046482, −6.10185806961238123220308828765, −4.94129822917354905007799934252, −4.08820193591867159270367213901, −3.25253865091576536278937424711, −1.47194441284852658314681539387, 1.28053369052907518824860796407, 2.65414269351717063673851599505, 3.60915012450795890228621062789, 4.49306891813830026134034218140, 5.77813585848633804144099652692, 6.33847382328587745131725544027, 7.26115564146905875525658041122, 8.609491177045263475144714356550, 9.415045423537118175917945632270, 10.30048997466344840454330423379

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