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Properties

Label 1-1033-1033.136-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.809 + 0.586i$
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  + (−0.985 + 0.169i)2-s + (0.658 − 0.752i)3-s + (0.942 − 0.334i)4-s + (−0.998 + 0.0608i)5-s + (−0.520 + 0.853i)6-s + (−0.639 + 0.768i)7-s + (−0.872 + 0.489i)8-s + (−0.133 − 0.991i)9-s + (0.973 − 0.229i)10-s + (0.987 + 0.157i)11-s + (0.368 − 0.929i)12-s + (0.991 − 0.133i)13-s + (0.5 − 0.866i)14-s + (−0.611 + 0.791i)15-s + (0.776 − 0.630i)16-s + (−0.435 − 0.900i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.169i)2-s + (0.658 − 0.752i)3-s + (0.942 − 0.334i)4-s + (−0.998 + 0.0608i)5-s + (−0.520 + 0.853i)6-s + (−0.639 + 0.768i)7-s + (−0.872 + 0.489i)8-s + (−0.133 − 0.991i)9-s + (0.973 − 0.229i)10-s + (0.987 + 0.157i)11-s + (0.368 − 0.929i)12-s + (0.991 − 0.133i)13-s + (0.5 − 0.866i)14-s + (−0.611 + 0.791i)15-s + (0.776 − 0.630i)16-s + (−0.435 − 0.900i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.809 + 0.586i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7901459866 + 0.2560869853i\)
\(L(\frac12)\) \(\approx\) \(0.7901459866 + 0.2560869853i\)
\(L(1)\) \(\approx\) \(0.7184620983 + 0.006873645536i\)
\(L(1)\) \(\approx\) \(0.7184620983 + 0.006873645536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.985 + 0.169i)T \)
3 \( 1 + (0.658 - 0.752i)T \)
5 \( 1 + (-0.998 + 0.0608i)T \)
7 \( 1 + (-0.639 + 0.768i)T \)
11 \( 1 + (0.987 + 0.157i)T \)
13 \( 1 + (0.991 - 0.133i)T \)
17 \( 1 + (-0.435 - 0.900i)T \)
19 \( 1 + (-0.276 + 0.961i)T \)
23 \( 1 + (-0.0243 + 0.999i)T \)
29 \( 1 + (-0.591 + 0.806i)T \)
31 \( 1 + (0.894 + 0.446i)T \)
37 \( 1 + (-0.934 - 0.357i)T \)
41 \( 1 + (-0.925 + 0.379i)T \)
43 \( 1 + (-0.205 - 0.978i)T \)
47 \( 1 + (-0.999 - 0.0121i)T \)
53 \( 1 + (0.551 + 0.833i)T \)
59 \( 1 + (0.591 + 0.806i)T \)
61 \( 1 + (-0.109 + 0.994i)T \)
67 \( 1 + (0.954 - 0.299i)T \)
71 \( 1 + (0.998 + 0.0608i)T \)
73 \( 1 + (0.357 + 0.934i)T \)
79 \( 1 + (-0.630 + 0.776i)T \)
83 \( 1 + (-0.0121 - 0.999i)T \)
89 \( 1 + (0.967 + 0.252i)T \)
97 \( 1 + (0.702 + 0.711i)T \)
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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

−21.17393839879572066546385109958, −20.42755252756987604131390129829, −19.86015231273521012735907561966, −19.2871487772855022070117536362, −18.78626509423424241994773248626, −17.348709371533676232858769468771, −16.763744339565656623627283517434, −16.056685973159874391789502006966, −15.438157463062681078747024953644, −14.7373527215432065199119072944, −13.57361868151553724980697984095, −12.73261985072233237394763145970, −11.49333165846021520418650828890, −11.01828373056210438684301522168, −10.19813509138450046635352153878, −9.36903825118432981655617143817, −8.49377548524627900191701726033, −8.15671109810329474505739849867, −6.897579842967615232557988724852, −6.3730012616164390678087227992, −4.527327597779883802732388057702, −3.747997875127827735009346042645, −3.2424445228833992850390114359, −1.90968653928283766475287169706, −0.542252964458216746676644255225, 1.01114463029647803057689016291, 1.97188993945642332240810246034, 3.16044561619606357085751766409, 3.71851982289306343192086028063, 5.53718850990107597248263900769, 6.59630006474996299758807531390, 7.00753697943141970606276339489, 8.022065014757482352894403065635, 8.74688886152710619185327468881, 9.174669716628990560790725829, 10.25836037380032643527642153500, 11.559181475828132472674680563455, 11.87004964490728460226203414714, 12.729052949549277881946957414499, 13.87314207047670797411281186369, 14.80780322307245520342043557497, 15.49435347056961052336873236607, 16.05879403218910306489123987848, 17.02371788695255128176429056419, 18.10417811033159824263506071145, 18.608654666972794153568462428504, 19.21790134919207344502440836948, 19.87138559833848577090004276687, 20.40220270046306498298991057407, 21.381102531434117556487389424343

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