Properties

Label 1-896-896.37-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.240 + 0.970i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (0.831 + 0.555i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.321 + 0.946i)37-s + ⋯
L(s)  = 1  + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (0.831 + 0.555i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)33-s + (0.321 + 0.946i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.240 + 0.970i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.240 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06836375156 + 0.08741294679i\)
\(L(\frac12)\) \(\approx\) \(0.06836375156 + 0.08741294679i\)
\(L(1)\) \(\approx\) \(0.5801486106 - 0.1675451064i\)
\(L(1)\) \(\approx\) \(0.5801486106 - 0.1675451064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.442 - 0.896i)T \)
5 \( 1 + (-0.946 + 0.321i)T \)
11 \( 1 + (-0.0654 - 0.997i)T \)
13 \( 1 + (-0.195 - 0.980i)T \)
17 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (-0.659 - 0.751i)T \)
23 \( 1 + (-0.793 - 0.608i)T \)
29 \( 1 + (0.831 + 0.555i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.321 + 0.946i)T \)
41 \( 1 + (-0.923 + 0.382i)T \)
43 \( 1 + (-0.555 - 0.831i)T \)
47 \( 1 + (0.965 + 0.258i)T \)
53 \( 1 + (0.0654 + 0.997i)T \)
59 \( 1 + (-0.751 - 0.659i)T \)
61 \( 1 + (-0.997 - 0.0654i)T \)
67 \( 1 + (0.442 + 0.896i)T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (-0.991 - 0.130i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + (-0.980 + 0.195i)T \)
89 \( 1 + (-0.130 - 0.991i)T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.65680031785386302513489218708, −20.972119012817875905675404029113, −20.19409492996695355538456227904, −19.57137739091021269633427492593, −18.50712251192054430792913989602, −17.669098302400361539794325511732, −16.66596153932455938321663865589, −16.239476724259354098834375308239, −15.41494288977282850646372875568, −14.74662297765934634671449959515, −13.8811600005825903863586391371, −12.43437732869850574093747990268, −12.00284249789939476772032897090, −11.26839437858378723324748007121, −10.268366767803591799107050289854, −9.51593544671876492507646342327, −8.72340428438471038927164652923, −7.65076922472254304502356533617, −6.8162699699023008906169582873, −5.65790244040055536230494679318, −4.63433649355587332745943653598, −4.19967937465919559770187815789, −3.20218547421473628833380740106, −1.76903503452464021224202196836, −0.05935458186896211995988288562, 1.08794790926805901373015992014, 2.516439646050579454867421793357, 3.36083189090868786094153997832, 4.531686090664591026937523001342, 5.66792980393322115543068917877, 6.4139794055300278394683296937, 7.30595119385678586632213034654, 8.20481339853474389854236419240, 8.578318200409259147227778299444, 10.37282608556113889477397539194, 10.8561274192906668281260142896, 11.73320922238409109053501537748, 12.47207011755178327983374046697, 13.16213939280220757671430446687, 14.12016687946483932911975328746, 15.00720387830486767364622879084, 15.81819431214998700768904079556, 16.761203875368549339259310491679, 17.38887927327572732901869124494, 18.55765190056649810008550855062, 18.76289131301622928365773591939, 19.84310256273118589996933153302, 20.159259748284621179083203966935, 21.81382391282356354376052223045, 22.10157920231901892506446064873

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