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Properties

Label 1-916-916.619-r1-0-0
Degree $1$
Conductor $916$
Sign $-0.745 - 0.666i$
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  + (−0.945 − 0.324i)3-s + (0.546 − 0.837i)5-s + (0.677 − 0.735i)7-s + (0.789 + 0.614i)9-s + (0.986 − 0.164i)11-s + (0.546 − 0.837i)13-s + (−0.789 + 0.614i)15-s + (0.546 − 0.837i)17-s + (−0.546 − 0.837i)19-s + (−0.879 + 0.475i)21-s + (0.0825 − 0.996i)23-s + (−0.401 − 0.915i)25-s + (−0.546 − 0.837i)27-s + (−0.677 + 0.735i)29-s + (0.986 − 0.164i)31-s + ⋯
L(s)  = 1  + (−0.945 − 0.324i)3-s + (0.546 − 0.837i)5-s + (0.677 − 0.735i)7-s + (0.789 + 0.614i)9-s + (0.986 − 0.164i)11-s + (0.546 − 0.837i)13-s + (−0.789 + 0.614i)15-s + (0.546 − 0.837i)17-s + (−0.546 − 0.837i)19-s + (−0.879 + 0.475i)21-s + (0.0825 − 0.996i)23-s + (−0.401 − 0.915i)25-s + (−0.546 − 0.837i)27-s + (−0.677 + 0.735i)29-s + (0.986 − 0.164i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ -0.745 - 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7946924888 - 2.079637306i\)
\(L(\frac12)\) \(\approx\) \(0.7946924888 - 2.079637306i\)
\(L(1)\) \(\approx\) \(0.9672354903 - 0.5790339957i\)
\(L(1)\) \(\approx\) \(0.9672354903 - 0.5790339957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.945 - 0.324i)T \)
5 \( 1 + (0.546 - 0.837i)T \)
7 \( 1 + (0.677 - 0.735i)T \)
11 \( 1 + (0.986 - 0.164i)T \)
13 \( 1 + (0.546 - 0.837i)T \)
17 \( 1 + (0.546 - 0.837i)T \)
19 \( 1 + (-0.546 - 0.837i)T \)
23 \( 1 + (0.0825 - 0.996i)T \)
29 \( 1 + (-0.677 + 0.735i)T \)
31 \( 1 + (0.986 - 0.164i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (0.789 - 0.614i)T \)
43 \( 1 + (0.879 - 0.475i)T \)
47 \( 1 + (-0.789 - 0.614i)T \)
53 \( 1 + (0.945 + 0.324i)T \)
59 \( 1 + (0.879 + 0.475i)T \)
61 \( 1 + (0.789 + 0.614i)T \)
67 \( 1 + (-0.789 - 0.614i)T \)
71 \( 1 + (0.986 + 0.164i)T \)
73 \( 1 + (-0.401 - 0.915i)T \)
79 \( 1 + (0.677 + 0.735i)T \)
83 \( 1 + (0.879 + 0.475i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.401 + 0.915i)T \)
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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

−21.96873708581712808381994963005, −21.151198526584576888948947113259, −21.07152064949530244477169117676, −19.187588840653071990330883397052, −18.913569964908531762678660888348, −17.75967320237843588820935296203, −17.466290301759144722969292419562, −16.57786883943718652441156199793, −15.612728442288816937315984461381, −14.73693405396402151931524229740, −14.30124199699221672241107897667, −13.08178240211550967559003110509, −12.02968541086411223149444990075, −11.48553760252723402954760283461, −10.782242658683508057676371278907, −9.835430578895367939010121048916, −9.17056361786683557675503736799, −7.98441004851460540632819914607, −6.81515422224358244235764451275, −6.09132658034769631632819998166, −5.57223762830858456018519152217, −4.30884252134615294733328860544, −3.56241194233038621178335385492, −2.00125708463936874731979302210, −1.33283517950716376634386506646, 0.68531566412206128628824796020, 1.01119128959090187085699676016, 2.21256420283421759506871537864, 3.867443176682344521112843021510, 4.77785790507099473081397344575, 5.41901150022241943253110257829, 6.39341570646730303683134831909, 7.18995513581672525375620178140, 8.236780353957446175852636364740, 9.07548875491633550252406681334, 10.21012734722188503641855270889, 10.842776551224918566536532866542, 11.75632402247764316252876046416, 12.4594764622573166099306874076, 13.37251887359353036244175461054, 13.90748323643114076245517187227, 15.01745204995085657858506519149, 16.23023777563225701294559367415, 16.69715544757122300832782487526, 17.51518524770680045561148660663, 17.8574786927855976196981892645, 18.934637317956769141265202582705, 19.92202846117406414621688079182, 20.72636213466356019316644733044, 21.28369262387178925676684301051

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