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Properties

Label 1-2557-2557.1061-r0-0-0
Degree $1$
Conductor $2557$
Sign $-0.246 + 0.969i$
Analytic cond. $11.8746$
Root an. cond. $11.8746$
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  + (0.805 − 0.592i)2-s + (0.0957 − 0.995i)3-s + (0.297 − 0.954i)4-s + (−0.461 − 0.887i)5-s + (−0.512 − 0.858i)6-s + (0.740 − 0.672i)7-s + (−0.325 − 0.945i)8-s + (−0.981 − 0.190i)9-s + (−0.897 − 0.441i)10-s + (−0.986 + 0.161i)11-s + (−0.921 − 0.387i)12-s + (0.978 − 0.205i)13-s + (0.197 − 0.980i)14-s + (−0.927 + 0.374i)15-s + (−0.822 − 0.568i)16-s + (0.688 − 0.725i)17-s + ⋯
L(s)  = 1  + (0.805 − 0.592i)2-s + (0.0957 − 0.995i)3-s + (0.297 − 0.954i)4-s + (−0.461 − 0.887i)5-s + (−0.512 − 0.858i)6-s + (0.740 − 0.672i)7-s + (−0.325 − 0.945i)8-s + (−0.981 − 0.190i)9-s + (−0.897 − 0.441i)10-s + (−0.986 + 0.161i)11-s + (−0.921 − 0.387i)12-s + (0.978 − 0.205i)13-s + (0.197 − 0.980i)14-s + (−0.927 + 0.374i)15-s + (−0.822 − 0.568i)16-s + (0.688 − 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $-0.246 + 0.969i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (1061, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ -0.246 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.271906506 - 1.635667288i\)
\(L(\frac12)\) \(\approx\) \(-1.271906506 - 1.635667288i\)
\(L(1)\) \(\approx\) \(0.6305458972 - 1.354346682i\)
\(L(1)\) \(\approx\) \(0.6305458972 - 1.354346682i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.805 - 0.592i)T \)
3 \( 1 + (0.0957 - 0.995i)T \)
5 \( 1 + (-0.461 - 0.887i)T \)
7 \( 1 + (0.740 - 0.672i)T \)
11 \( 1 + (-0.986 + 0.161i)T \)
13 \( 1 + (0.978 - 0.205i)T \)
17 \( 1 + (0.688 - 0.725i)T \)
19 \( 1 + (-0.139 - 0.990i)T \)
23 \( 1 + (0.00737 + 0.999i)T \)
29 \( 1 + (-0.999 - 0.0147i)T \)
31 \( 1 + (0.854 + 0.519i)T \)
37 \( 1 + (0.212 - 0.977i)T \)
41 \( 1 + (-0.956 + 0.290i)T \)
43 \( 1 + (0.997 + 0.0736i)T \)
47 \( 1 + (-0.978 - 0.205i)T \)
53 \( 1 + (-0.787 - 0.616i)T \)
59 \( 1 + (-0.968 + 0.248i)T \)
61 \( 1 + (0.699 + 0.714i)T \)
67 \( 1 + (0.574 - 0.818i)T \)
71 \( 1 + (-0.474 - 0.880i)T \)
73 \( 1 + (-0.550 + 0.835i)T \)
79 \( 1 + (0.999 + 0.0442i)T \)
83 \( 1 + (-0.998 - 0.0589i)T \)
89 \( 1 + (0.968 - 0.248i)T \)
97 \( 1 + (-0.367 - 0.930i)T \)
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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

−20.456111254298998609970501226685, −18.90676707110430182784775528876, −18.61678090666129486087255493261, −17.62768897292422720515387199198, −16.80413877058883185512653826396, −16.09379759351874230727529143109, −15.47688973935338545973597075138, −14.995704297342950821737740342696, −14.41623823329533446564302253441, −13.84106290575086105708868989467, −12.7921518146208174554110080833, −11.9442635789026162468206237251, −11.24051233955535958421794712160, −10.73973107048317310014743967293, −9.92729117858364988444153637393, −8.59922197983375430222498547134, −8.1971830959903912379054979122, −7.628193914173382345401263138752, −6.23576722916451579694917592142, −5.92282972543232145572176351231, −5.00345716007701344862679042149, −4.24351201249373494978925534664, −3.500358806779807816996972774779, −2.86718589344212225844152218787, −1.97146017751647439430393176030, 0.48379469348741593380875525711, 1.23008455050936982561905629287, 1.95723005399891006617786637739, 3.04862651592018851652396489620, 3.76153174309576173232471945882, 4.868697218556752824870460082046, 5.257732990243409277080446966715, 6.15752157328028667653234701456, 7.293087135488528085190933255993, 7.69830346771039845287549020593, 8.587916999312304765759918307178, 9.44099684992124895110307037130, 10.52138159851199452953829069160, 11.393698647490577795910982710399, 11.59589305952323007173050254174, 12.657341611815884992364199674, 13.14198085659723168711931247966, 13.635212869369182295234646236804, 14.30513660142289538825558575798, 15.29048097041701582604813942861, 15.876451821385419286688083661267, 16.7706055734043288395221381736, 17.74541263212694739437983710339, 18.23752301131083865059069378031, 19.12043836432022246551198087215

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