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Properties

Label 1-1096-1096.221-r1-0-0
Degree $1$
Conductor $1096$
Sign $0.925 - 0.378i$
Analytic cond. $117.781$
Root an. cond. $117.781$
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  + (0.973 − 0.228i)3-s + (0.0461 + 0.998i)5-s + (−0.961 + 0.273i)7-s + (0.895 − 0.445i)9-s + (0.995 + 0.0922i)11-s + (0.873 + 0.486i)13-s + (0.273 + 0.961i)15-s + (−0.798 − 0.602i)17-s + (0.361 − 0.932i)19-s + (−0.873 + 0.486i)21-s + (0.824 − 0.565i)23-s + (−0.995 + 0.0922i)25-s + (0.769 − 0.638i)27-s + (−0.565 − 0.824i)29-s + (0.403 − 0.914i)31-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)3-s + (0.0461 + 0.998i)5-s + (−0.961 + 0.273i)7-s + (0.895 − 0.445i)9-s + (0.995 + 0.0922i)11-s + (0.873 + 0.486i)13-s + (0.273 + 0.961i)15-s + (−0.798 − 0.602i)17-s + (0.361 − 0.932i)19-s + (−0.873 + 0.486i)21-s + (0.824 − 0.565i)23-s + (−0.995 + 0.0922i)25-s + (0.769 − 0.638i)27-s + (−0.565 − 0.824i)29-s + (0.403 − 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(117.781\)
Root analytic conductor: \(117.781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1096,\ (1:\ ),\ 0.925 - 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.172636829 - 0.6234763004i\)
\(L(\frac12)\) \(\approx\) \(3.172636829 - 0.6234763004i\)
\(L(1)\) \(\approx\) \(1.552473100 + 0.001981018620i\)
\(L(1)\) \(\approx\) \(1.552473100 + 0.001981018620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
137 \( 1 \)
good3 \( 1 + (0.973 - 0.228i)T \)
5 \( 1 + (0.0461 + 0.998i)T \)
7 \( 1 + (-0.961 + 0.273i)T \)
11 \( 1 + (0.995 + 0.0922i)T \)
13 \( 1 + (0.873 + 0.486i)T \)
17 \( 1 + (-0.798 - 0.602i)T \)
19 \( 1 + (0.361 - 0.932i)T \)
23 \( 1 + (0.824 - 0.565i)T \)
29 \( 1 + (-0.565 - 0.824i)T \)
31 \( 1 + (0.403 - 0.914i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.914 + 0.403i)T \)
47 \( 1 + (0.317 - 0.948i)T \)
53 \( 1 + (0.403 + 0.914i)T \)
59 \( 1 + (-0.445 - 0.895i)T \)
61 \( 1 + (-0.895 - 0.445i)T \)
67 \( 1 + (0.486 - 0.873i)T \)
71 \( 1 + (0.769 + 0.638i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.228 + 0.973i)T \)
83 \( 1 + (0.990 + 0.138i)T \)
89 \( 1 + (-0.998 + 0.0461i)T \)
97 \( 1 + (0.638 + 0.769i)T \)
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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

−21.038913392492698656668981790695, −20.44956089018496059167982901735, −19.716085408835597665807234213111, −19.33749774417254200511977399890, −18.29974562553793222987195460561, −17.19309303729290490300671123478, −16.48972210808494012634765259187, −15.82759878705814563091903878481, −15.12841249133938686030534759044, −14.08267714656630890753958916376, −13.36511071556438033792646863992, −12.85968576234068881024002349748, −12.00123112851980825800146971473, −10.74032581903200525792177122479, −9.94170689349163386681385948226, −9.00230743696249804253334435404, −8.76416173093408975547099318159, −7.70755755513510155719413468743, −6.7084903672309850991519838245, −5.79185518748409058393349366136, −4.625288291503198862409260399807, −3.71557536901065540465468044622, −3.23715377868166365436867447376, −1.719318730768087928916034793296, −1.01112182360396051205512509822, 0.65177557090268254949752946539, 2.062406465621036685042746913963, 2.786090617127490837757452139260, 3.597772676874990316320667635218, 4.372639149615189734701608795314, 6.05651331516112757421418883507, 6.72650283641840080460440529351, 7.199806875016377461674325545542, 8.426333586220202269184080592482, 9.38827753954245321131336340919, 9.57239123738929579064451076100, 10.935743134884282832268157065630, 11.58698607152267227791318404373, 12.703753947543730307458136297651, 13.56394777628224074550965280970, 13.94896987219063970494953450432, 15.04921315201134886549937292319, 15.419079797099604926100244472905, 16.35008197869347580444791177883, 17.42985833245852074328625074514, 18.527577837147017663255830305177, 18.71531797113763390778556450593, 19.69693969022030131172137058116, 20.102490951625992965416386664, 21.26364764598488331275137521027

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