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Properties

Label 2-848-212.143-c0-0-0
Degree $2$
Conductor $848$
Sign $0.638 + 0.769i$
Analytic cond. $0.423207$
Root an. cond. $0.650543$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.869 − 0.329i)5-s + (0.354 − 0.935i)9-s + (0.213 − 0.112i)13-s + (1.45 − 1.28i)17-s + (−0.101 − 0.0902i)25-s + (0.0854 − 0.704i)29-s + (0.850 + 1.23i)37-s + (−1.31 + 0.159i)41-s + (−0.616 + 0.695i)45-s + (0.885 + 0.464i)49-s + (−0.568 − 0.822i)53-s + (−1.24 + 1.39i)61-s + (−0.222 + 0.0270i)65-s + (1.09 + 1.23i)73-s + (−0.748 − 0.663i)81-s + ⋯
L(s)  = 1  + (−0.869 − 0.329i)5-s + (0.354 − 0.935i)9-s + (0.213 − 0.112i)13-s + (1.45 − 1.28i)17-s + (−0.101 − 0.0902i)25-s + (0.0854 − 0.704i)29-s + (0.850 + 1.23i)37-s + (−1.31 + 0.159i)41-s + (−0.616 + 0.695i)45-s + (0.885 + 0.464i)49-s + (−0.568 − 0.822i)53-s + (−1.24 + 1.39i)61-s + (−0.222 + 0.0270i)65-s + (1.09 + 1.23i)73-s + (−0.748 − 0.663i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(848\)    =    \(2^{4} \cdot 53\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(0.423207\)
Root analytic conductor: \(0.650543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{848} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 848,\ (\ :0),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8752980454\)
\(L(\frac12)\) \(\approx\) \(0.8752980454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
53 \( 1 + (0.568 + 0.822i)T \)
good3 \( 1 + (-0.354 + 0.935i)T^{2} \)
5 \( 1 + (0.869 + 0.329i)T + (0.748 + 0.663i)T^{2} \)
7 \( 1 + (-0.885 - 0.464i)T^{2} \)
11 \( 1 + (0.970 - 0.239i)T^{2} \)
13 \( 1 + (-0.213 + 0.112i)T + (0.568 - 0.822i)T^{2} \)
17 \( 1 + (-1.45 + 1.28i)T + (0.120 - 0.992i)T^{2} \)
19 \( 1 + (0.568 - 0.822i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.0854 + 0.704i)T + (-0.970 - 0.239i)T^{2} \)
31 \( 1 + (-0.970 - 0.239i)T^{2} \)
37 \( 1 + (-0.850 - 1.23i)T + (-0.354 + 0.935i)T^{2} \)
41 \( 1 + (1.31 - 0.159i)T + (0.970 - 0.239i)T^{2} \)
43 \( 1 + (0.354 + 0.935i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
59 \( 1 + (0.748 - 0.663i)T^{2} \)
61 \( 1 + (1.24 - 1.39i)T + (-0.120 - 0.992i)T^{2} \)
67 \( 1 + (0.568 + 0.822i)T^{2} \)
71 \( 1 + (-0.354 - 0.935i)T^{2} \)
73 \( 1 + (-1.09 - 1.23i)T + (-0.120 + 0.992i)T^{2} \)
79 \( 1 + (0.885 - 0.464i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.12 - 0.992i)T + (0.120 - 0.992i)T^{2} \)
97 \( 1 + (0.251 - 0.663i)T + (-0.748 - 0.663i)T^{2} \)
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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

−10.05445173840467564620341414323, −9.561008610711056942800282899590, −8.494201901890698869390966569348, −7.76453513677304088492935608752, −6.95359483551933804302489372891, −5.92127570108007526984514821646, −4.82896415695533935372620066430, −3.87841713882613420151407430742, −2.96778221645688441423434739457, −1.00530400678433386647928721262, 1.72124871556899464390110899817, 3.25998016055265053412069370742, 4.07608322116015739548470649927, 5.18914794154091013173205548889, 6.17008099634301661852611298056, 7.37131254484353491092991594340, 7.82599297506609514646595434297, 8.654414252104686702116543000179, 9.826632784596983975728727974038, 10.62235807130440967555513668653

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