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Properties

Label 1-2e9-512.53-r0-0-0
Degree $1$
Conductor $512$
Sign $0.460 + 0.887i$
Analytic cond. $2.37771$
Root an. cond. $2.37771$
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  + (−0.671 + 0.740i)3-s + (0.970 + 0.242i)5-s + (0.773 − 0.634i)7-s + (−0.0980 − 0.995i)9-s + (−0.427 + 0.903i)11-s + (−0.514 + 0.857i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.803 − 0.595i)19-s + (−0.0490 + 0.998i)21-s + (−0.956 − 0.290i)23-s + (0.881 + 0.471i)25-s + (0.803 + 0.595i)27-s + (0.336 − 0.941i)29-s + (−0.382 + 0.923i)31-s + ⋯
L(s)  = 1  + (−0.671 + 0.740i)3-s + (0.970 + 0.242i)5-s + (0.773 − 0.634i)7-s + (−0.0980 − 0.995i)9-s + (−0.427 + 0.903i)11-s + (−0.514 + 0.857i)13-s + (−0.831 + 0.555i)15-s + (0.831 + 0.555i)17-s + (0.803 − 0.595i)19-s + (−0.0490 + 0.998i)21-s + (−0.956 − 0.290i)23-s + (0.881 + 0.471i)25-s + (0.803 + 0.595i)27-s + (0.336 − 0.941i)29-s + (−0.382 + 0.923i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(2.37771\)
Root analytic conductor: \(2.37771\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 512,\ (0:\ ),\ 0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169800123 + 0.7109437973i\)
\(L(\frac12)\) \(\approx\) \(1.169800123 + 0.7109437973i\)
\(L(1)\) \(\approx\) \(1.044958669 + 0.3220602301i\)
\(L(1)\) \(\approx\) \(1.044958669 + 0.3220602301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.671 + 0.740i)T \)
5 \( 1 + (0.970 + 0.242i)T \)
7 \( 1 + (0.773 - 0.634i)T \)
11 \( 1 + (-0.427 + 0.903i)T \)
13 \( 1 + (-0.514 + 0.857i)T \)
17 \( 1 + (0.831 + 0.555i)T \)
19 \( 1 + (0.803 - 0.595i)T \)
23 \( 1 + (-0.956 - 0.290i)T \)
29 \( 1 + (0.336 - 0.941i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (0.989 - 0.146i)T \)
41 \( 1 + (0.881 - 0.471i)T \)
43 \( 1 + (-0.740 + 0.671i)T \)
47 \( 1 + (-0.195 - 0.980i)T \)
53 \( 1 + (0.336 + 0.941i)T \)
59 \( 1 + (0.514 + 0.857i)T \)
61 \( 1 + (0.0490 + 0.998i)T \)
67 \( 1 + (-0.998 + 0.0490i)T \)
71 \( 1 + (-0.995 - 0.0980i)T \)
73 \( 1 + (0.773 + 0.634i)T \)
79 \( 1 + (0.980 + 0.195i)T \)
83 \( 1 + (0.989 + 0.146i)T \)
89 \( 1 + (-0.956 + 0.290i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−23.67753933801928263577344230619, −22.41074143276344449078136504519, −21.91610896044978019596436690523, −21.025876062138671082856536813, −20.14907314278156148901896138521, −18.88684899557718651988077598946, −18.14015783053698222120333579683, −17.795086374848051276983838911509, −16.69487819601179032771610993875, −16.07253419541625476823472561348, −14.61330434260269654881151742812, −13.916942736798720315774690079489, −13.022735579462874948057249680490, −12.21202465797917428759126271686, −11.40351753320080886121897880439, −10.42366135944965152099819250966, −9.4904727192019272235912843034, −8.16985132175867897852749249258, −7.64985730930116011246995987921, −6.165445305591981114297525936775, −5.53004623540397006579907112987, −4.998599531093023522747520453836, −3.00972354616459021026671988077, −1.99585084502382396322124773023, −0.92675586304239397942345352572, 1.29918185016668184763441681231, 2.50615982154721299732061965282, 4.028822782475792947047068127094, 4.84017288812689189082004552874, 5.662327658869981409400894335721, 6.73346416164682696318543699378, 7.66608167112840759840013973358, 9.10771287400397727317235491037, 10.00849888965134188821028327983, 10.41662296042889896906719392041, 11.51413689871643229974565093971, 12.31361412490509882415801394326, 13.55490473169007162445260077876, 14.4378763906023031829986102749, 15.033679242778571565702838647674, 16.310760332185090960261045987050, 16.93925443744474895722501381210, 17.8460957206994635636421488830, 18.13762181656418803301793357847, 19.70838459594434994531721694234, 20.65735731726078983504345337408, 21.31325689790889715701249495634, 21.88149244413785455059575007969, 22.86990333886266964331996008012, 23.61134951614308744490241541304

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