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Properties

Label 2-848-212.7-c0-0-0
Degree $2$
Conductor $848$
Sign $0.327 + 0.944i$
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  + (0.922 − 1.75i)5-s + (−0.885 − 0.464i)9-s + (−0.0854 − 0.704i)13-s + (−0.850 + 1.23i)17-s + (−1.67 − 2.42i)25-s + (0.627 + 1.65i)29-s + (1.10 + 0.271i)37-s + (1.53 + 0.583i)41-s + (−1.63 + 1.12i)45-s + (0.120 − 0.992i)49-s + (0.970 + 0.239i)53-s + (0.764 − 0.527i)61-s + (−1.31 − 0.499i)65-s + (0.393 + 0.271i)73-s + (0.568 + 0.822i)81-s + ⋯
L(s)  = 1  + (0.922 − 1.75i)5-s + (−0.885 − 0.464i)9-s + (−0.0854 − 0.704i)13-s + (−0.850 + 1.23i)17-s + (−1.67 − 2.42i)25-s + (0.627 + 1.65i)29-s + (1.10 + 0.271i)37-s + (1.53 + 0.583i)41-s + (−1.63 + 1.12i)45-s + (0.120 − 0.992i)49-s + (0.970 + 0.239i)53-s + (0.764 − 0.527i)61-s + (−1.31 − 0.499i)65-s + (0.393 + 0.271i)73-s + (0.568 + 0.822i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(848\)    =    \(2^{4} \cdot 53\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(0.423207\)
Root analytic conductor: \(0.650543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{848} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 848,\ (\ :0),\ 0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.041788924\)
\(L(\frac12)\) \(\approx\) \(1.041788924\)
\(L(1)\) 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 \( 1 + (-0.970 - 0.239i)T \)
good3 \( 1 + (0.885 + 0.464i)T^{2} \)
5 \( 1 + (-0.922 + 1.75i)T + (-0.568 - 0.822i)T^{2} \)
7 \( 1 + (-0.120 + 0.992i)T^{2} \)
11 \( 1 + (0.748 + 0.663i)T^{2} \)
13 \( 1 + (0.0854 + 0.704i)T + (-0.970 + 0.239i)T^{2} \)
17 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
19 \( 1 + (-0.970 + 0.239i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.627 - 1.65i)T + (-0.748 + 0.663i)T^{2} \)
31 \( 1 + (-0.748 + 0.663i)T^{2} \)
37 \( 1 + (-1.10 - 0.271i)T + (0.885 + 0.464i)T^{2} \)
41 \( 1 + (-1.53 - 0.583i)T + (0.748 + 0.663i)T^{2} \)
43 \( 1 + (-0.885 + 0.464i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
59 \( 1 + (-0.568 + 0.822i)T^{2} \)
61 \( 1 + (-0.764 + 0.527i)T + (0.354 - 0.935i)T^{2} \)
67 \( 1 + (-0.970 - 0.239i)T^{2} \)
71 \( 1 + (0.885 - 0.464i)T^{2} \)
73 \( 1 + (-0.393 - 0.271i)T + (0.354 + 0.935i)T^{2} \)
79 \( 1 + (0.120 + 0.992i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.645 - 0.935i)T + (-0.354 - 0.935i)T^{2} \)
97 \( 1 + (1.56 + 0.822i)T + (0.568 + 0.822i)T^{2} \)
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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

−10.07704105270304424244137473783, −9.221470388651002179095279029746, −8.631059796902836662371241697432, −8.077357908829187861389074109169, −6.49112375140511277941388232409, −5.73946657776939737305262778236, −5.04893233183037593148168850199, −4.01297645467105394946558738952, −2.48758395344283905216480427170, −1.13075655490348370700991644027, 2.38798330698158103084472794467, 2.68419295837919624781604710366, 4.18636111198383672374183926732, 5.54525459115056679453371025665, 6.27266539727602631992009732727, 7.01869105466699342708031925955, 7.83490375424070639746708100264, 9.141375074474084049272606103027, 9.718056218473614692163107170679, 10.65467765177878321669243873262

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