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Properties

Label 1-1033-1033.1017-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.456 - 0.889i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (−0.950 − 0.311i)2-s + (0.0121 − 0.999i)3-s + (0.806 + 0.591i)4-s + (0.847 − 0.531i)5-s + (−0.322 + 0.946i)6-s + (0.744 + 0.667i)7-s + (−0.581 − 0.813i)8-s + (−0.999 − 0.0243i)9-s + (−0.970 + 0.241i)10-s + (−0.676 − 0.736i)11-s + (0.601 − 0.798i)12-s + (0.999 + 0.0243i)13-s + (−0.5 − 0.866i)14-s + (−0.520 − 0.853i)15-s + (0.299 + 0.954i)16-s + (0.0608 + 0.998i)17-s + ⋯
L(s)  = 1  + (−0.950 − 0.311i)2-s + (0.0121 − 0.999i)3-s + (0.806 + 0.591i)4-s + (0.847 − 0.531i)5-s + (−0.322 + 0.946i)6-s + (0.744 + 0.667i)7-s + (−0.581 − 0.813i)8-s + (−0.999 − 0.0243i)9-s + (−0.970 + 0.241i)10-s + (−0.676 − 0.736i)11-s + (0.601 − 0.798i)12-s + (0.999 + 0.0243i)13-s + (−0.5 − 0.866i)14-s + (−0.520 − 0.853i)15-s + (0.299 + 0.954i)16-s + (0.0608 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (1017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.456 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.081546572 - 0.6604188871i\)
\(L(\frac12)\) \(\approx\) \(1.081546572 - 0.6604188871i\)
\(L(1)\) \(\approx\) \(0.8435347152 - 0.3644519240i\)
\(L(1)\) \(\approx\) \(0.8435347152 - 0.3644519240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.950 - 0.311i)T \)
3 \( 1 + (0.0121 - 0.999i)T \)
5 \( 1 + (0.847 - 0.531i)T \)
7 \( 1 + (0.744 + 0.667i)T \)
11 \( 1 + (-0.676 - 0.736i)T \)
13 \( 1 + (0.999 + 0.0243i)T \)
17 \( 1 + (0.0608 + 0.998i)T \)
19 \( 1 + (-0.368 + 0.929i)T \)
23 \( 1 + (0.658 + 0.752i)T \)
29 \( 1 + (0.985 + 0.169i)T \)
31 \( 1 + (0.883 + 0.468i)T \)
37 \( 1 + (-0.976 - 0.217i)T \)
41 \( 1 + (0.478 - 0.877i)T \)
43 \( 1 + (-0.820 - 0.571i)T \)
47 \( 1 + (0.413 - 0.910i)T \)
53 \( 1 + (0.0365 + 0.999i)T \)
59 \( 1 + (0.985 - 0.169i)T \)
61 \( 1 + (0.639 - 0.768i)T \)
67 \( 1 + (-0.942 - 0.334i)T \)
71 \( 1 + (0.847 + 0.531i)T \)
73 \( 1 + (0.976 - 0.217i)T \)
79 \( 1 + (-0.299 + 0.954i)T \)
83 \( 1 + (-0.413 - 0.910i)T \)
89 \( 1 + (0.457 + 0.889i)T \)
97 \( 1 + (0.541 + 0.840i)T \)
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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

−21.18634191773880194358753127771, −20.97592866359141264202478764635, −20.35024624219672816373933721563, −19.348619990934565441763750746746, −18.23387438792396388868448382033, −17.76593988220086023934501186937, −17.15835906447128638078232463657, −16.28049550602167888491556945785, −15.519069014799451595453788624403, −14.82225760795206467515325425268, −14.094556359045734696828401342981, −13.28805671095901419451926154635, −11.61110328198086715372517013528, −10.99121500187432664332299627021, −10.336767868583269121101168205572, −9.81364635062803573807329488220, −8.87113284714539421570935327445, −8.13445798742512548610531482707, −7.04838011350557816187619141458, −6.32147455286238775286348315989, −5.18779917488240001116528632781, −4.590586064114247080060456637408, −3.00951259920389886845414392902, −2.29427500395980558648603017642, −0.94465468527031196353766618862, 1.01351495893710373527787162557, 1.68826422724108744786578512381, 2.454520515510816408054675709988, 3.55408352187208710866441379766, 5.34969789432998656060282926325, 5.949554308802049221174108536976, 6.77684611431840924019018072663, 8.11216705989879036850731550074, 8.41522731573415016849471716176, 9.00612563367503330048951036650, 10.35322773569684723251952268035, 10.93282374552107851118365394383, 11.96217733582703065056077765239, 12.53246005812813988366719424647, 13.39122024697434993684984239152, 14.10843187602733675765630167783, 15.31351230934228220740068874826, 16.189346703878340903053025047700, 17.22464245916415413210706176321, 17.52778023084624307823245124691, 18.47014278938901018286518073312, 18.80311783557699207268063763387, 19.6839405948737609089887482195, 20.76009130302247163395461222304, 21.16019270356229919075333036317

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