Properties

Label 2-848-212.131-c0-0-0
Degree $2$
Conductor $848$
Sign $-0.502 + 0.864i$
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  + (−1.85 + 0.225i)5-s + (−0.120 − 0.992i)9-s + (−0.627 − 1.65i)13-s + (−1.10 − 0.271i)17-s + (2.42 − 0.597i)25-s + (−0.213 − 0.112i)29-s + (−1.45 − 1.28i)37-s + (−0.222 − 0.423i)41-s + (0.447 + 1.81i)45-s + (−0.354 + 0.935i)49-s + (0.748 + 0.663i)53-s + (0.475 + 1.92i)61-s + (1.53 + 2.93i)65-s + (0.317 − 1.28i)73-s + (−0.970 + 0.239i)81-s + ⋯
L(s)  = 1  + (−1.85 + 0.225i)5-s + (−0.120 − 0.992i)9-s + (−0.627 − 1.65i)13-s + (−1.10 − 0.271i)17-s + (2.42 − 0.597i)25-s + (−0.213 − 0.112i)29-s + (−1.45 − 1.28i)37-s + (−0.222 − 0.423i)41-s + (0.447 + 1.81i)45-s + (−0.354 + 0.935i)49-s + (0.748 + 0.663i)53-s + (0.475 + 1.92i)61-s + (1.53 + 2.93i)65-s + (0.317 − 1.28i)73-s + (−0.970 + 0.239i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(848\)    =    \(2^{4} \cdot 53\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(0.423207\)
Root analytic conductor: \(0.650543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{848} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 848,\ (\ :0),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4046322602\)
\(L(\frac12)\) \(\approx\) \(0.4046322602\)
\(L(1)\) 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 \( 1 + (-0.748 - 0.663i)T \)
good3 \( 1 + (0.120 + 0.992i)T^{2} \)
5 \( 1 + (1.85 - 0.225i)T + (0.970 - 0.239i)T^{2} \)
7 \( 1 + (0.354 - 0.935i)T^{2} \)
11 \( 1 + (-0.568 + 0.822i)T^{2} \)
13 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
17 \( 1 + (1.10 + 0.271i)T + (0.885 + 0.464i)T^{2} \)
19 \( 1 + (-0.748 + 0.663i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.213 + 0.112i)T + (0.568 + 0.822i)T^{2} \)
31 \( 1 + (0.568 + 0.822i)T^{2} \)
37 \( 1 + (1.45 + 1.28i)T + (0.120 + 0.992i)T^{2} \)
41 \( 1 + (0.222 + 0.423i)T + (-0.568 + 0.822i)T^{2} \)
43 \( 1 + (-0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.970 + 0.239i)T^{2} \)
59 \( 1 + (0.970 + 0.239i)T^{2} \)
61 \( 1 + (-0.475 - 1.92i)T + (-0.885 + 0.464i)T^{2} \)
67 \( 1 + (-0.748 - 0.663i)T^{2} \)
71 \( 1 + (0.120 - 0.992i)T^{2} \)
73 \( 1 + (-0.317 + 1.28i)T + (-0.885 - 0.464i)T^{2} \)
79 \( 1 + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1.88 + 0.464i)T + (0.885 + 0.464i)T^{2} \)
97 \( 1 + (0.0290 + 0.239i)T + (-0.970 + 0.239i)T^{2} \)
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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

−10.31921564292485189680651478060, −9.059834361723879670296640541025, −8.398726654569444856464313729746, −7.46477609433770895067927534866, −6.98989341600602442030049243244, −5.70496771328448425076599586664, −4.53375840842495456711590861751, −3.65960505907243030133662323204, −2.84442528667955493269856791381, −0.39702639747643795056937830584, 2.03932453743227596279404872008, 3.52041574099702163260689005504, 4.46606486456958764062452716969, 4.98947830711861773129513340764, 6.74574109921501036133918787686, 7.21132487711206135910752098417, 8.294644062742697583822232346941, 8.643871802958718155588102798652, 9.836224028375933745323607973470, 10.99115702526924994973457121992

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