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Properties

Label 1-896-896.557-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: $\chi_{896} (557, \cdot )$
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

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

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