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Properties

Label 2-848-212.59-c0-0-0
Degree $2$
Conductor $848$
Sign $0.899 - 0.436i$
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  + (1.09 + 1.23i)5-s + (0.748 − 0.663i)9-s + (−1.10 − 1.59i)13-s + (0.213 + 1.75i)17-s + (−0.206 + 1.69i)25-s + (−1.45 + 0.358i)29-s + (0.0854 + 0.225i)37-s + (0.475 − 1.92i)41-s + (1.63 + 0.198i)45-s + (0.568 − 0.822i)49-s + (0.354 + 0.935i)53-s + (−1.31 − 0.159i)61-s + (0.764 − 3.10i)65-s + (−1.85 + 0.225i)73-s + (0.120 − 0.992i)81-s + ⋯
L(s)  = 1  + (1.09 + 1.23i)5-s + (0.748 − 0.663i)9-s + (−1.10 − 1.59i)13-s + (0.213 + 1.75i)17-s + (−0.206 + 1.69i)25-s + (−1.45 + 0.358i)29-s + (0.0854 + 0.225i)37-s + (0.475 − 1.92i)41-s + (1.63 + 0.198i)45-s + (0.568 − 0.822i)49-s + (0.354 + 0.935i)53-s + (−1.31 − 0.159i)61-s + (0.764 − 3.10i)65-s + (−1.85 + 0.225i)73-s + (0.120 − 0.992i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(848\)    =    \(2^{4} \cdot 53\)
Sign: $0.899 - 0.436i$
Analytic conductor: \(0.423207\)
Root analytic conductor: \(0.650543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{848} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 848,\ (\ :0),\ 0.899 - 0.436i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173044141\)
\(L(\frac12)\) \(\approx\) \(1.173044141\)
\(L(1)\) 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 \( 1 + (-0.354 - 0.935i)T \)
good3 \( 1 + (-0.748 + 0.663i)T^{2} \)
5 \( 1 + (-1.09 - 1.23i)T + (-0.120 + 0.992i)T^{2} \)
7 \( 1 + (-0.568 + 0.822i)T^{2} \)
11 \( 1 + (-0.885 - 0.464i)T^{2} \)
13 \( 1 + (1.10 + 1.59i)T + (-0.354 + 0.935i)T^{2} \)
17 \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \)
19 \( 1 + (-0.354 + 0.935i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (1.45 - 0.358i)T + (0.885 - 0.464i)T^{2} \)
31 \( 1 + (0.885 - 0.464i)T^{2} \)
37 \( 1 + (-0.0854 - 0.225i)T + (-0.748 + 0.663i)T^{2} \)
41 \( 1 + (-0.475 + 1.92i)T + (-0.885 - 0.464i)T^{2} \)
43 \( 1 + (0.748 + 0.663i)T^{2} \)
47 \( 1 + (-0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.120 - 0.992i)T^{2} \)
61 \( 1 + (1.31 + 0.159i)T + (0.970 + 0.239i)T^{2} \)
67 \( 1 + (-0.354 - 0.935i)T^{2} \)
71 \( 1 + (-0.748 - 0.663i)T^{2} \)
73 \( 1 + (1.85 - 0.225i)T + (0.970 - 0.239i)T^{2} \)
79 \( 1 + (0.568 + 0.822i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.0290 + 0.239i)T + (-0.970 + 0.239i)T^{2} \)
97 \( 1 + (1.12 - 0.992i)T + (0.120 - 0.992i)T^{2} \)
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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

−10.35017517410391486475146041792, −9.912001793769333106710013464347, −8.944395950867312765484366111006, −7.65075450077340097252557919970, −7.06060963056008236303911551452, −6.02208774842043587761808911646, −5.49970101592030489689825760173, −3.91813790318598989947538379942, −2.94744379066041130216556104233, −1.81591409748974254322808158760, 1.53501299269810506006333529670, 2.46925831298911345086039285025, 4.44524250263036527385994970615, 4.85638624232301352193393835954, 5.80655404006093963519694236274, 6.99604394762635150008711580162, 7.69037843887185839760011085062, 8.982594968206131106948396208747, 9.530929047582916878539068923903, 9.913438687867214591951945372298

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