Properties

Label 1-916-916.511-r1-0-0
Degree $1$
Conductor $916$
Sign $-0.636 + 0.770i$
Analytic cond. $98.4378$
Root an. cond. $98.4378$
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  + (0.879 − 0.475i)3-s + (−0.0825 − 0.996i)5-s + (−0.945 + 0.324i)7-s + (0.546 − 0.837i)9-s + (−0.245 + 0.969i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 − 0.837i)15-s + (−0.0825 − 0.996i)17-s + (0.0825 − 0.996i)19-s + (−0.677 + 0.735i)21-s + (−0.789 − 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.0825 − 0.996i)27-s + (0.945 − 0.324i)29-s + (−0.245 + 0.969i)31-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)3-s + (−0.0825 − 0.996i)5-s + (−0.945 + 0.324i)7-s + (0.546 − 0.837i)9-s + (−0.245 + 0.969i)11-s + (−0.0825 − 0.996i)13-s + (−0.546 − 0.837i)15-s + (−0.0825 − 0.996i)17-s + (0.0825 − 0.996i)19-s + (−0.677 + 0.735i)21-s + (−0.789 − 0.614i)23-s + (−0.986 + 0.164i)25-s + (0.0825 − 0.996i)27-s + (0.945 − 0.324i)29-s + (−0.245 + 0.969i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $-0.636 + 0.770i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ -0.636 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2443923969 - 0.5189637882i\)
\(L(\frac12)\) \(\approx\) \(-0.2443923969 - 0.5189637882i\)
\(L(1)\) \(\approx\) \(0.9541521556 - 0.4439122559i\)
\(L(1)\) \(\approx\) \(0.9541521556 - 0.4439122559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (0.879 - 0.475i)T \)
5 \( 1 + (-0.0825 - 0.996i)T \)
7 \( 1 + (-0.945 + 0.324i)T \)
11 \( 1 + (-0.245 + 0.969i)T \)
13 \( 1 + (-0.0825 - 0.996i)T \)
17 \( 1 + (-0.0825 - 0.996i)T \)
19 \( 1 + (0.0825 - 0.996i)T \)
23 \( 1 + (-0.789 - 0.614i)T \)
29 \( 1 + (0.945 - 0.324i)T \)
31 \( 1 + (-0.245 + 0.969i)T \)
37 \( 1 + (-0.677 + 0.735i)T \)
41 \( 1 + (0.546 + 0.837i)T \)
43 \( 1 + (0.677 - 0.735i)T \)
47 \( 1 + (-0.546 + 0.837i)T \)
53 \( 1 + (-0.879 + 0.475i)T \)
59 \( 1 + (0.677 + 0.735i)T \)
61 \( 1 + (0.546 - 0.837i)T \)
67 \( 1 + (-0.546 + 0.837i)T \)
71 \( 1 + (-0.245 - 0.969i)T \)
73 \( 1 + (-0.986 + 0.164i)T \)
79 \( 1 + (-0.945 - 0.324i)T \)
83 \( 1 + (0.677 + 0.735i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.986 - 0.164i)T \)
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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

−21.96303827154749266587580483150, −21.56713908201214744764815985391, −20.68265643689368971403769612552, −19.50837713810215096524352565175, −19.249611652898388049186512272560, −18.62741030921452985807641803946, −17.4357291496334658731128033930, −16.16592565082317012658037626819, −16.07685177084763438665183122358, −14.891763304219091325013159163120, −14.1938012287008123576653227342, −13.67518150468236627198517270352, −12.73832778938935693396729503478, −11.55534890065789306037704954220, −10.59112297771143742499564587002, −10.073011551209005456294809200898, −9.22610513387054247723977246252, −8.2515548540967894870691242304, −7.4487476369101934249916929305, −6.49322947515697373914878278247, −5.70674625677846763970131746596, −4.024830891264941767223181739723, −3.678225633021611592041381798141, −2.74980141951083367867022709126, −1.75898153528051284690233477145, 0.10766436822159621932611194013, 1.120331796946490626837309152541, 2.42607689750394161685220963501, 3.05540673496342762146731530283, 4.313115293263150879511088385180, 5.14782733131152080063225092996, 6.35528115604925987420767103795, 7.2342601713717352270789841323, 8.069377712746213288013381790512, 8.91218255872063971998020783954, 9.569154895336926812581971922834, 10.29278297656692608417202586941, 11.931991468727213861203095439751, 12.47230679745080797297230958078, 13.08986777876220588833882277154, 13.76606834630499106974642176570, 14.85160710498093912982511392816, 15.836736893464591375958917108167, 15.97507933336792436089403493418, 17.523749892321973258973687470374, 17.93195168041254570389554139115, 19.01861199313971660678896450349, 19.79656370707553761345348274581, 20.2630671813189302418927604037, 20.86035902645936212358690432258

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