Properties

Label 1-1045-1045.237-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.106 - 0.994i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.829 − 0.559i)2-s + (−0.139 − 0.990i)3-s + (0.374 + 0.927i)4-s + (−0.438 + 0.898i)6-s + (0.207 + 0.978i)7-s + (0.207 − 0.978i)8-s + (−0.961 + 0.275i)9-s + (0.866 − 0.5i)12-s + (0.694 − 0.719i)13-s + (0.374 − 0.927i)14-s + (−0.719 + 0.694i)16-s + (0.275 − 0.961i)17-s + (0.951 + 0.309i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.997 − 0.0697i)24-s + ⋯
L(s)  = 1  + (−0.829 − 0.559i)2-s + (−0.139 − 0.990i)3-s + (0.374 + 0.927i)4-s + (−0.438 + 0.898i)6-s + (0.207 + 0.978i)7-s + (0.207 − 0.978i)8-s + (−0.961 + 0.275i)9-s + (0.866 − 0.5i)12-s + (0.694 − 0.719i)13-s + (0.374 − 0.927i)14-s + (−0.719 + 0.694i)16-s + (0.275 − 0.961i)17-s + (0.951 + 0.309i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (−0.997 − 0.0697i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.106 - 0.994i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.106 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5989838663 - 0.6666092875i\)
\(L(\frac12)\) \(\approx\) \(0.5989838663 - 0.6666092875i\)
\(L(1)\) \(\approx\) \(0.6410372998 - 0.3432330939i\)
\(L(1)\) \(\approx\) \(0.6410372998 - 0.3432330939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.829 - 0.559i)T \)
3 \( 1 + (-0.139 - 0.990i)T \)
7 \( 1 + (0.207 + 0.978i)T \)
13 \( 1 + (0.694 - 0.719i)T \)
17 \( 1 + (0.275 - 0.961i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (-0.615 + 0.788i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.990 + 0.139i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (0.999 + 0.0348i)T \)
53 \( 1 + (0.970 + 0.241i)T \)
59 \( 1 + (-0.0348 - 0.999i)T \)
61 \( 1 + (0.997 - 0.0697i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (0.469 - 0.882i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.829 + 0.559i)T \)
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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

−21.59835614695819659272320306927, −20.8328031564957988764834940956, −20.18470047457764309444235374416, −19.43942079147330015881250606774, −18.5675957687195770756784580876, −17.52096414427823966495132218072, −17.00605124599989358970879431834, −16.430367119747097952080510189781, −15.60241646619180866281907761480, −14.93719817760463897624312472052, −14.09050450829889392737139224647, −13.42005056155213843996386544351, −11.705046779658814333154061312113, −11.22144176407457337844897151922, −10.21511059275194081816718770258, −9.923248493667254556651361923729, −8.81814174295016332602368684835, −8.17313500740908505861987683303, −7.204611282332183313419889013204, −6.19941871263128526462027447819, −5.51355226138052551613068081685, −4.31596548263694237452822860270, −3.72384435136149732988252747265, −2.142519558340335984473745514013, −0.929468174910056686111788370227, 0.66717535736034555429360682830, 1.71434724497792838555174482925, 2.60642825680011984537272693143, 3.32596972249598509550275858202, 4.924837109025997017688977293012, 5.9853988511196745393388358885, 6.77286260647659808945241122101, 7.81747217234277182746251452519, 8.36757628131626451055651179969, 9.10022527113380004540435531458, 10.14831655380143450089536334890, 11.13342772899619576992342505858, 11.77191781543059109689414023727, 12.42922962151742753492262177689, 13.129570094391143421079377464631, 14.044198308976972737933286845357, 15.17215685071121010704438881020, 16.08455893501559171286677343540, 16.85846839848001334598514800960, 17.8163368317967565149430822052, 18.495971796743664922091916700, 18.55528608378567286072738082607, 19.75547214342572778902137668655, 20.326043491714770610343306946794, 21.12524677623476869154843444300

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