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Properties

Label 1-919-919.416-r0-0-0
Degree $1$
Conductor $919$
Sign $-0.880 - 0.473i$
Analytic cond. $4.26781$
Root an. cond. $4.26781$
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  + (0.932 − 0.361i)2-s + (0.445 − 0.895i)3-s + (0.739 − 0.673i)4-s + (0.445 − 0.895i)5-s + (0.0922 − 0.995i)6-s + (0.0922 − 0.995i)7-s + (0.445 − 0.895i)8-s + (−0.602 − 0.798i)9-s + (0.0922 − 0.995i)10-s + (−0.602 + 0.798i)11-s + (−0.273 − 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯
L(s)  = 1  + (0.932 − 0.361i)2-s + (0.445 − 0.895i)3-s + (0.739 − 0.673i)4-s + (0.445 − 0.895i)5-s + (0.0922 − 0.995i)6-s + (0.0922 − 0.995i)7-s + (0.445 − 0.895i)8-s + (−0.602 − 0.798i)9-s + (0.0922 − 0.995i)10-s + (−0.602 + 0.798i)11-s + (−0.273 − 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (416, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ -0.880 - 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7964507528 - 3.164779353i\)
\(L(\frac12)\) \(\approx\) \(0.7964507528 - 3.164779353i\)
\(L(1)\) \(\approx\) \(1.501859822 - 1.581709635i\)
\(L(1)\) \(\approx\) \(1.501859822 - 1.581709635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad919 \( 1 \)
good2 \( 1 + (0.932 - 0.361i)T \)
3 \( 1 + (0.445 - 0.895i)T \)
5 \( 1 + (0.445 - 0.895i)T \)
7 \( 1 + (0.0922 - 0.995i)T \)
11 \( 1 + (-0.602 + 0.798i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
17 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (0.445 + 0.895i)T \)
23 \( 1 + (-0.602 + 0.798i)T \)
29 \( 1 + (0.932 - 0.361i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (0.739 + 0.673i)T \)
41 \( 1 + (-0.273 + 0.961i)T \)
43 \( 1 + (0.445 + 0.895i)T \)
47 \( 1 + (0.739 - 0.673i)T \)
53 \( 1 + T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (-0.982 + 0.183i)T \)
71 \( 1 + (0.0922 - 0.995i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
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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

−22.01202824447161632698773143213, −21.699938955709610777435808417657, −20.98355640881974205472366688190, −20.125079885419642376101228813778, −19.19843273970880162361143094927, −18.11826508090544861295066603734, −17.46683319273325870974631632797, −16.08494353739971662581337163550, −15.73526107277085275675356833026, −15.09959338829302999024780666441, −14.25982169931639397041915658483, −13.67099796131778183175046388610, −12.84613269411904152152791243034, −11.645709440488343289072023185629, −10.80178220590229133426177554419, −10.39826100161340565898089443053, −8.84320718055087149765568919586, −8.45218052823395539656972089854, −7.25917797762092572822814650759, −6.06112180119094633515608264914, −5.64774587852749451792857703771, −4.65537547292291325800690934624, −3.52792065478387674011497707456, −2.79919797300751038118393939580, −2.23907177604948498452914362966, 0.9872706712559047676299441622, 1.72781713215639191602516957977, 2.61909846552189969848072237277, 3.93081957823840250962201186984, 4.55724625359926148821018575020, 5.7160596014020970885426989469, 6.51206328984039719417954822227, 7.41723999251925249182302512147, 8.21113178498467135423043836361, 9.490215507179628008332361352479, 10.11746443907118858309146892773, 11.41853786249265948593542078185, 12.00285063570402993708783546280, 13.006686791606176056650130958130, 13.506250873776021417215296101291, 13.883300896368459486480582188222, 14.90444040768533222774983412348, 15.9292450401692594505113957708, 16.68997732087690915276283324741, 17.757354885753064869760249481532, 18.41437194870837527186842199463, 19.58856851105002016137990647846, 20.143644618229698994255352799127, 20.70507402915647372324562837695, 21.231700729636116378643340851900

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