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Properties

Label 1-2e10-1024.37-r0-0-0
Degree $1$
Conductor $1024$
Sign $0.560 - 0.828i$
Analytic cond. $4.75543$
Root an. cond. $4.75543$
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  + (0.870 − 0.492i)3-s + (0.817 + 0.575i)5-s + (−0.146 − 0.989i)7-s + (0.514 − 0.857i)9-s + (0.313 − 0.949i)11-s + (−0.534 + 0.844i)13-s + (0.995 + 0.0980i)15-s + (0.0980 + 0.995i)17-s + (−0.0245 − 0.999i)19-s + (−0.615 − 0.788i)21-s + (0.671 − 0.740i)23-s + (0.336 + 0.941i)25-s + (0.0245 − 0.999i)27-s + (0.997 + 0.0735i)29-s + (−0.195 + 0.980i)31-s + ⋯
L(s)  = 1  + (0.870 − 0.492i)3-s + (0.817 + 0.575i)5-s + (−0.146 − 0.989i)7-s + (0.514 − 0.857i)9-s + (0.313 − 0.949i)11-s + (−0.534 + 0.844i)13-s + (0.995 + 0.0980i)15-s + (0.0980 + 0.995i)17-s + (−0.0245 − 0.999i)19-s + (−0.615 − 0.788i)21-s + (0.671 − 0.740i)23-s + (0.336 + 0.941i)25-s + (0.0245 − 0.999i)27-s + (0.997 + 0.0735i)29-s + (−0.195 + 0.980i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1024\)    =    \(2^{10}\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(4.75543\)
Root analytic conductor: \(4.75543\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1024} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1024,\ (0:\ ),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.137990882 - 1.134360443i\)
\(L(\frac12)\) \(\approx\) \(2.137990882 - 1.134360443i\)
\(L(1)\) \(\approx\) \(1.573726295 - 0.4091421127i\)
\(L(1)\) \(\approx\) \(1.573726295 - 0.4091421127i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.870 - 0.492i)T \)
5 \( 1 + (0.817 + 0.575i)T \)
7 \( 1 + (-0.146 - 0.989i)T \)
11 \( 1 + (0.313 - 0.949i)T \)
13 \( 1 + (-0.534 + 0.844i)T \)
17 \( 1 + (0.0980 + 0.995i)T \)
19 \( 1 + (-0.0245 - 0.999i)T \)
23 \( 1 + (0.671 - 0.740i)T \)
29 \( 1 + (0.997 + 0.0735i)T \)
31 \( 1 + (-0.195 + 0.980i)T \)
37 \( 1 + (-0.932 + 0.359i)T \)
41 \( 1 + (0.336 - 0.941i)T \)
43 \( 1 + (0.963 - 0.266i)T \)
47 \( 1 + (0.290 - 0.956i)T \)
53 \( 1 + (-0.997 + 0.0735i)T \)
59 \( 1 + (-0.844 + 0.534i)T \)
61 \( 1 + (0.788 + 0.615i)T \)
67 \( 1 + (-0.122 - 0.992i)T \)
71 \( 1 + (0.242 - 0.970i)T \)
73 \( 1 + (0.146 - 0.989i)T \)
79 \( 1 + (-0.881 - 0.471i)T \)
83 \( 1 + (-0.932 - 0.359i)T \)
89 \( 1 + (0.671 + 0.740i)T \)
97 \( 1 + (0.555 + 0.831i)T \)
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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

−21.602612041826881092177199671549, −20.81395489846179688224019091972, −20.39123563060517338836128607416, −19.48992277150546097564255613151, −18.674400572401076630035155645322, −17.76574145592479437621362407510, −17.07935005512197277012912555377, −15.976436535247792942234867912922, −15.49217783984727047723857349164, −14.52762238610754863462953873537, −14.04325806995981341622850249822, −12.78036874487356176978730457859, −12.58958406464804635042616022850, −11.34576793991278966572225483814, −9.929305958910922067928963924191, −9.74994847051822411152365261610, −8.9590342919244936093089375655, −8.09734467297094805747434907810, −7.21309399425551841935742171044, −5.90309395088025799990902403976, −5.14833200743874414348705936911, −4.40414575375675200208315305182, −3.05301261578497500448953271397, −2.392723272849514450388908264893, −1.43747146816981642659501301426, 0.98052678083609973619520837406, 1.980550885852954879819576169, 2.95789618344884339281490226305, 3.7131448684292326847549493750, 4.81836612627981647800007341923, 6.28044987158745141319909107117, 6.758905806514153551290152097604, 7.48692040735042943757959414976, 8.71451522620310286993997425443, 9.17912842620863358966819089384, 10.33586770625138944813277031646, 10.79760482067330286839090109256, 12.070912150745695298984673522404, 13.00263263261288613174271161195, 13.794583248991442574832787400945, 14.13417530719384638770276296127, 14.86031360767260029120287464030, 15.9541731699075251416163018243, 17.052190381254390859336747895862, 17.46806359235174068413488896704, 18.52212274415038049220897472260, 19.32747066386805128664998471744, 19.61301942163505225267637219226, 20.76929432655389491545268797949, 21.410043179318583247538024407

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