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Properties

Label 1-1049-1049.122-r0-0-0
Degree $1$
Conductor $1049$
Sign $-0.668 + 0.743i$
Analytic cond. $4.87153$
Root an. cond. $4.87153$
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  + (0.811 − 0.583i)2-s + (−0.995 + 0.0957i)3-s + (0.318 − 0.948i)4-s + (−0.999 − 0.0239i)5-s + (−0.752 + 0.658i)6-s + (−0.782 − 0.622i)7-s + (−0.295 − 0.955i)8-s + (0.981 − 0.190i)9-s + (−0.825 + 0.564i)10-s + (−0.736 − 0.676i)11-s + (−0.225 + 0.974i)12-s + (0.649 − 0.760i)13-s + (−0.998 − 0.0479i)14-s + (0.997 − 0.0718i)15-s + (−0.797 − 0.603i)16-s + (0.935 − 0.352i)17-s + ⋯
L(s)  = 1  + (0.811 − 0.583i)2-s + (−0.995 + 0.0957i)3-s + (0.318 − 0.948i)4-s + (−0.999 − 0.0239i)5-s + (−0.752 + 0.658i)6-s + (−0.782 − 0.622i)7-s + (−0.295 − 0.955i)8-s + (0.981 − 0.190i)9-s + (−0.825 + 0.564i)10-s + (−0.736 − 0.676i)11-s + (−0.225 + 0.974i)12-s + (0.649 − 0.760i)13-s + (−0.998 − 0.0479i)14-s + (0.997 − 0.0718i)15-s + (−0.797 − 0.603i)16-s + (0.935 − 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.668 + 0.743i$
Analytic conductor: \(4.87153\)
Root analytic conductor: \(4.87153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (0:\ ),\ -0.668 + 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2887345021 - 0.6477544132i\)
\(L(\frac12)\) \(\approx\) \(-0.2887345021 - 0.6477544132i\)
\(L(1)\) \(\approx\) \(0.6208912997 - 0.5693825027i\)
\(L(1)\) \(\approx\) \(0.6208912997 - 0.5693825027i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (0.811 - 0.583i)T \)
3 \( 1 + (-0.995 + 0.0957i)T \)
5 \( 1 + (-0.999 - 0.0239i)T \)
7 \( 1 + (-0.782 - 0.622i)T \)
11 \( 1 + (-0.736 - 0.676i)T \)
13 \( 1 + (0.649 - 0.760i)T \)
17 \( 1 + (0.935 - 0.352i)T \)
19 \( 1 + (-0.429 - 0.903i)T \)
23 \( 1 + (0.976 + 0.214i)T \)
29 \( 1 + (-0.107 - 0.994i)T \)
31 \( 1 + (-0.534 - 0.845i)T \)
37 \( 1 + (-0.958 + 0.283i)T \)
41 \( 1 + (0.429 + 0.903i)T \)
43 \( 1 + (0.0359 - 0.999i)T \)
47 \( 1 + (-0.981 + 0.190i)T \)
53 \( 1 + (-0.385 + 0.922i)T \)
59 \( 1 + (0.998 - 0.0479i)T \)
61 \( 1 + (-0.554 + 0.832i)T \)
67 \( 1 + (-0.908 + 0.418i)T \)
71 \( 1 + (-0.752 - 0.658i)T \)
73 \( 1 + (0.944 - 0.329i)T \)
79 \( 1 + (-0.554 - 0.832i)T \)
83 \( 1 + (0.985 + 0.167i)T \)
89 \( 1 + (-0.782 + 0.622i)T \)
97 \( 1 + (-0.0119 + 0.999i)T \)
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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

−22.44204374618900734656933632366, −21.23984799864095469185537214843, −20.98630574065970197147963841351, −19.59234540219278531106011704221, −18.75667394648520378828888406721, −18.15871305571249536154805171032, −17.017215625611275199491190131489, −16.19795279458463447881335152415, −16.01565427886802313818276101105, −15.09534119937108743906671779285, −14.35457080782555201390933544675, −12.8756585571769464935685232924, −12.66827906821585516965567506476, −11.980861814806918497593965115158, −11.128017933663810313646159501163, −10.30972958736428550478617952493, −8.96802262719957548834843901484, −8.02340162546963720698442628112, −7.09008638917487461456531843135, −6.57088734348758877111292512692, −5.55938476890710029123081666654, −4.91798444114800981072202468396, −3.890829306603098610525493604250, −3.18091982131170433981268206410, −1.72299174169757908957036067800, 0.306753457100684055348206826588, 1.04762591815343623888205868477, 2.9048256164882773868432770289, 3.54389971877131920931715826211, 4.41738016738680733863801953408, 5.304358607884216689439566671220, 6.07569968758884306491660243466, 6.98963507689079247824659397876, 7.79277905783067469580971480026, 9.24036560493625580951410168929, 10.3086771391803654496743423269, 10.82405351411364392339333494227, 11.45439219789050589708790235959, 12.27770775494122624832672524012, 13.154776667543967773588695120917, 13.43268395895662286057444596471, 14.94479736213948971443607169119, 15.58182169237833139410392898838, 16.17323078471571145286962242375, 16.88970005625784338563116551206, 18.129096259676427135117768268433, 19.02098264537400559076153240835, 19.33798527231581775746105338137, 20.54966912550018428699144808067, 20.965978388093465211614730926216

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