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Properties

Label 1-896-896.339-r0-0-0
Degree $1$
Conductor $896$
Sign $0.891 + 0.453i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445098673 + 0.3462675206i\)
\(L(\frac12)\) \(\approx\) \(1.445098673 + 0.3462675206i\)
\(L(1)\) \(\approx\) \(1.082196431 + 0.2307043224i\)
\(L(1)\) \(\approx\) \(1.082196431 + 0.2307043224i\)

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

−21.95228423845950252646638770774, −21.14805435034169404113276309048, −20.26992179703873449727310866482, −19.484034194662204350528473747617, −18.52813330945459152801836310756, −17.92262359641660852859388939974, −17.22087003688383733954821331534, −16.64128068112213292013311218471, −15.586816657137059822084131101830, −14.40546993124947709375752390871, −13.748403027148522837738592739636, −13.00211581803766361791636482127, −12.35616540716412195523249602671, −11.414036798574669640627801466959, −10.64924573578703032565399058810, −9.39819999166160894008591477454, −8.980352725993666154651933761707, −7.62816094937360960625092660645, −6.846839107657488817367542300283, −6.22902497746185353756171966056, −5.120094274635102656541745711845, −4.5028501598801690074797709951, −2.70411922595573801614684962860, −1.9724939777511467889529225053, −1.0240037791341686381112755650, 0.88164197977064448312627383599, 2.39155882244821300518987323957, 3.36226230991992006571029784944, 4.2541557803908280241089084663, 5.586343892633298566544872344176, 5.836696280748972908718183202, 6.80110078423839240289462514077, 8.31365439436423234294054848232, 8.930213890649334079118590118752, 9.969994326155865751562396882455, 10.700896106704321682272792843163, 11.013165162909693872084888483, 12.40491756529954188959364625759, 13.106560046170624059252800902032, 14.22075990226999217336120859117, 14.765869955773769159196093404467, 15.66973301324934016556743055305, 16.54552518956809285811476678393, 17.207647166474242628827577709229, 17.84751926168079655693924595603, 18.75102320390235902702967844634, 19.7127948793214773385034929329, 20.75667196952006393574657132831, 21.33821818202232552996571532359, 21.871054653907969645488660571930

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