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Properties

Label 1-1033-1033.24-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.212 + 0.977i$
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  + (0.806 + 0.591i)2-s + (0.999 + 0.0243i)3-s + (0.299 + 0.954i)4-s + (0.900 + 0.435i)5-s + (0.791 + 0.611i)6-s + (−0.109 − 0.994i)7-s + (−0.322 + 0.946i)8-s + (0.998 + 0.0486i)9-s + (0.468 + 0.883i)10-s + (−0.996 − 0.0851i)11-s + (0.276 + 0.961i)12-s + (−0.0486 + 0.998i)13-s + (0.5 − 0.866i)14-s + (0.889 + 0.457i)15-s + (−0.820 + 0.571i)16-s + (0.992 − 0.121i)17-s + ⋯
L(s)  = 1  + (0.806 + 0.591i)2-s + (0.999 + 0.0243i)3-s + (0.299 + 0.954i)4-s + (0.900 + 0.435i)5-s + (0.791 + 0.611i)6-s + (−0.109 − 0.994i)7-s + (−0.322 + 0.946i)8-s + (0.998 + 0.0486i)9-s + (0.468 + 0.883i)10-s + (−0.996 − 0.0851i)11-s + (0.276 + 0.961i)12-s + (−0.0486 + 0.998i)13-s + (0.5 − 0.866i)14-s + (0.889 + 0.457i)15-s + (−0.820 + 0.571i)16-s + (0.992 − 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.051409709 + 2.458160931i\)
\(L(\frac12)\) \(\approx\) \(3.051409709 + 2.458160931i\)
\(L(1)\) \(\approx\) \(2.233416738 + 1.082801787i\)
\(L(1)\) \(\approx\) \(2.233416738 + 1.082801787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.806 + 0.591i)T \)
3 \( 1 + (0.999 + 0.0243i)T \)
5 \( 1 + (0.900 + 0.435i)T \)
7 \( 1 + (-0.109 - 0.994i)T \)
11 \( 1 + (-0.996 - 0.0851i)T \)
13 \( 1 + (-0.0486 + 0.998i)T \)
17 \( 1 + (0.992 - 0.121i)T \)
19 \( 1 + (-0.728 - 0.685i)T \)
23 \( 1 + (0.991 + 0.133i)T \)
29 \( 1 + (-0.334 + 0.942i)T \)
31 \( 1 + (-0.561 - 0.827i)T \)
37 \( 1 + (0.905 + 0.424i)T \)
41 \( 1 + (0.541 + 0.840i)T \)
43 \( 1 + (-0.345 - 0.938i)T \)
47 \( 1 + (-0.752 + 0.658i)T \)
53 \( 1 + (-0.0729 - 0.997i)T \)
59 \( 1 + (0.334 + 0.942i)T \)
61 \( 1 + (0.181 + 0.983i)T \)
67 \( 1 + (0.630 - 0.776i)T \)
71 \( 1 + (-0.900 + 0.435i)T \)
73 \( 1 + (-0.424 - 0.905i)T \)
79 \( 1 + (0.571 - 0.820i)T \)
83 \( 1 + (0.658 - 0.752i)T \)
89 \( 1 + (-0.813 - 0.581i)T \)
97 \( 1 + (-0.910 - 0.413i)T \)
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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

−21.24554074745257185758970626921, −20.85734397936824619891191476943, −20.11935306285692501103697005649, −19.1055012007944311299379786713, −18.59899031114314868380818612978, −17.8151336944036571257650163266, −16.434667348760384266652466589568, −15.583412728512608916064698465381, −14.86896198237630824561686677515, −14.33128816727866872687950105891, −13.26826555866132000291327204072, −12.76854376704636685914558961742, −12.392904757400483575250309418491, −10.91284823026913248450642212338, −10.07298514983962289956418112402, −9.54928070476712145228508345214, −8.58255955692335774407097161138, −7.73803452575435032964826811064, −6.387155553734685546827968799044, −5.508805568151459269713931397012, −4.983509692201278852181355284411, −3.678532610055069684338800554755, −2.72772733538623112223562709290, −2.263487750951901331821427782086, −1.197980034245534253334424636502, 1.621716497008070816373026603587, 2.66134279646675404905866575335, 3.31064147847177035059189380200, 4.34738556851291727972470055638, 5.15448291571923174264644488891, 6.327640440383310798866042294976, 7.13728969986635440757430723766, 7.63088499926700701150256748879, 8.74309252799891034084695558535, 9.59670570742970658731284838563, 10.50631128096909645375463130216, 11.34331236860625071429189634322, 12.90859788519575695376439982201, 13.1641747963335150458952267402, 13.885289556850489909998792596879, 14.62436647014320066003038891066, 15.055678249230520369598955835625, 16.34414289007035769456330822624, 16.7038450886941779562945039494, 17.78473330829847993918744548130, 18.612646925915179976449186731436, 19.482786034786605586897776027187, 20.587258615610715289147050722000, 21.03581593179637759900946422522, 21.59104902968900906010969685494

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