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Properties

Label 1-2527-2527.989-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.988 + 0.154i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (0.0275 + 0.999i)2-s + (0.350 − 0.936i)3-s + (−0.998 + 0.0550i)4-s + (0.993 − 0.110i)5-s + (0.945 + 0.324i)6-s + (−0.0825 − 0.996i)8-s + (−0.754 − 0.656i)9-s + (0.137 + 0.990i)10-s + (−0.754 + 0.656i)11-s + (−0.298 + 0.954i)12-s + (−0.986 + 0.164i)13-s + (0.245 − 0.969i)15-s + (0.993 − 0.110i)16-s + (0.451 + 0.892i)17-s + (0.635 − 0.771i)18-s + ⋯
L(s)  = 1  + (0.0275 + 0.999i)2-s + (0.350 − 0.936i)3-s + (−0.998 + 0.0550i)4-s + (0.993 − 0.110i)5-s + (0.945 + 0.324i)6-s + (−0.0825 − 0.996i)8-s + (−0.754 − 0.656i)9-s + (0.137 + 0.990i)10-s + (−0.754 + 0.656i)11-s + (−0.298 + 0.954i)12-s + (−0.986 + 0.164i)13-s + (0.245 − 0.969i)15-s + (0.993 − 0.110i)16-s + (0.451 + 0.892i)17-s + (0.635 − 0.771i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.988 + 0.154i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.988 + 0.154i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769885693 + 0.1372198898i\)
\(L(\frac12)\) \(\approx\) \(1.769885693 + 0.1372198898i\)
\(L(1)\) \(\approx\) \(1.174753625 + 0.1919219543i\)
\(L(1)\) \(\approx\) \(1.174753625 + 0.1919219543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0275 + 0.999i)T \)
3 \( 1 + (0.350 - 0.936i)T \)
5 \( 1 + (0.993 - 0.110i)T \)
11 \( 1 + (-0.754 + 0.656i)T \)
13 \( 1 + (-0.986 + 0.164i)T \)
17 \( 1 + (0.451 + 0.892i)T \)
23 \( 1 + (0.350 + 0.936i)T \)
29 \( 1 + (0.546 - 0.837i)T \)
31 \( 1 + (0.0275 - 0.999i)T \)
37 \( 1 + (-0.191 - 0.981i)T \)
41 \( 1 + (0.245 - 0.969i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (-0.191 - 0.981i)T \)
53 \( 1 + (-0.754 + 0.656i)T \)
59 \( 1 + (0.716 + 0.697i)T \)
61 \( 1 + (0.904 - 0.426i)T \)
67 \( 1 + (0.904 + 0.426i)T \)
71 \( 1 + (-0.0825 - 0.996i)T \)
73 \( 1 + (0.451 + 0.892i)T \)
79 \( 1 + (0.137 - 0.990i)T \)
83 \( 1 + (-0.401 - 0.915i)T \)
89 \( 1 + (0.451 - 0.892i)T \)
97 \( 1 + (-0.0825 + 0.996i)T \)
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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

−19.50627093482565454931419722256, −18.83591043661118907690673675467, −18.08003537893380285003423381171, −17.38784335850789934473431459363, −16.63078583909226906749375494223, −15.95873533737837716133619727766, −14.82348537333317186697639654720, −14.24529907833396213696182052663, −13.82941571163001246276672435300, −12.921506285617242577908768125253, −12.27587103078465393530327891977, −11.14403817418377202996071123531, −10.71063215292733360383655073753, −9.86262327016826650926925436523, −9.65150024740619331741898473670, −8.66127787056733729084128253900, −8.12482910704747247806842138990, −6.84833815154890429428653676838, −5.59033638241737919698976713842, −5.09059091276968092077084384442, −4.53644390262938470737597493470, −3.13844123004394662656120322684, −2.91207639719001887871239005769, −2.110768304304477962886358902282, −0.83087477808723061106903813987, 0.70269128341311736580792838788, 1.88413997355189577838343700617, 2.53864283508542325744323612775, 3.71390893642942074425117794719, 4.80891583040841973911683361239, 5.61157569141501823418505621969, 6.10245603126179460551739433276, 7.06476386181871123295201636399, 7.558329117780768128848213133075, 8.26447104297766469495772629069, 9.18882861631150110739351516034, 9.72022861133491300515520302550, 10.47942081238004070492802658153, 11.85964534001637645606633434027, 12.71586545624274816316049627410, 13.03343920044428064721661535927, 13.79884779842202589826846544566, 14.46657744551330615251903624357, 14.98482535969520764395126640903, 15.821483657397225024119196103632, 16.8987147486387005280446741429, 17.517319875244936840869275246056, 17.64127416751789376614999892794, 18.693325550565810434427371694976, 19.135654243245092159551514021599

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