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Properties

Label 1-1021-1021.141-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.999 - 0.0403i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.343 + 0.938i)2-s + (0.932 − 0.361i)3-s + (−0.763 + 0.645i)4-s + (−0.320 + 0.947i)5-s + (0.659 + 0.751i)6-s + (−0.956 − 0.291i)7-s + (−0.869 − 0.494i)8-s + (0.739 − 0.673i)9-s + (−0.999 + 0.0246i)10-s + (0.923 − 0.384i)11-s + (−0.478 + 0.878i)12-s + (0.273 + 0.961i)13-s + (−0.0554 − 0.998i)14-s + (0.0431 + 0.999i)15-s + (0.165 − 0.986i)16-s + (−0.678 + 0.734i)17-s + ⋯
L(s)  = 1  + (0.343 + 0.938i)2-s + (0.932 − 0.361i)3-s + (−0.763 + 0.645i)4-s + (−0.320 + 0.947i)5-s + (0.659 + 0.751i)6-s + (−0.956 − 0.291i)7-s + (−0.869 − 0.494i)8-s + (0.739 − 0.673i)9-s + (−0.999 + 0.0246i)10-s + (0.923 − 0.384i)11-s + (−0.478 + 0.878i)12-s + (0.273 + 0.961i)13-s + (−0.0554 − 0.998i)14-s + (0.0431 + 0.999i)15-s + (0.165 − 0.986i)16-s + (−0.678 + 0.734i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.999 - 0.0403i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.999 - 0.0403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02647034024 + 1.311106819i\)
\(L(\frac12)\) \(\approx\) \(0.02647034024 + 1.311106819i\)
\(L(1)\) \(\approx\) \(0.9125278133 + 0.7801755206i\)
\(L(1)\) \(\approx\) \(0.9125278133 + 0.7801755206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.343 + 0.938i)T \)
3 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (-0.320 + 0.947i)T \)
7 \( 1 + (-0.956 - 0.291i)T \)
11 \( 1 + (0.923 - 0.384i)T \)
13 \( 1 + (0.273 + 0.961i)T \)
17 \( 1 + (-0.678 + 0.734i)T \)
19 \( 1 + (-0.213 + 0.976i)T \)
23 \( 1 + (-0.936 - 0.349i)T \)
29 \( 1 + (-0.225 + 0.974i)T \)
31 \( 1 + (-0.401 + 0.916i)T \)
37 \( 1 + (-0.355 + 0.934i)T \)
41 \( 1 + (0.213 - 0.976i)T \)
43 \( 1 + (-0.856 + 0.515i)T \)
47 \( 1 + (0.892 + 0.451i)T \)
53 \( 1 + (-0.963 + 0.267i)T \)
59 \( 1 + (-0.261 - 0.965i)T \)
61 \( 1 + (-0.836 + 0.547i)T \)
67 \( 1 + (-0.978 - 0.207i)T \)
71 \( 1 + (0.869 + 0.494i)T \)
73 \( 1 + (-0.0554 + 0.998i)T \)
79 \( 1 + (0.903 - 0.429i)T \)
83 \( 1 + (-0.153 + 0.988i)T \)
89 \( 1 + (-0.0307 - 0.999i)T \)
97 \( 1 + (0.389 - 0.920i)T \)
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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

−21.0254275973477647938184790786, −20.19342566351693146783425881729, −19.9018128922272459871678631930, −19.36004228090852966570730569219, −18.36649388029101301217820456835, −17.40585521558936215566685012570, −16.296175595413375030756835622361, −15.42894553297151599708637144926, −15.034631008713359821903073634158, −13.642971350808042624565709995, −13.37483692473692116034198046416, −12.52361400698107493177567310602, −11.80534637061529684064778787315, −10.77537947098609953192125098095, −9.63029526089360971839659330170, −9.35398225283547336605372192842, −8.62479128356825652116613183129, −7.59926553988988756481154482255, −6.24146716999662844822039067374, −5.17225326025554225653647883570, −4.249051371351435826050686369862, −3.675793991530478228542054261383, −2.6942467916100230831856497064, −1.80258564034104142372000602220, −0.421004004467994221220587594634, 1.67332002473654771614302936625, 3.04887799583674536500541257105, 3.73697169803543088789988950967, 4.21531875556577594582039719851, 6.09876422180649007007738651010, 6.57730441924742205570922847757, 7.115128336166930900918768139631, 8.14395311566649133496065321951, 8.872665931287398332497448209491, 9.6560603996302082408455488803, 10.668400512881423806437747970749, 12.03006028864094690713983936733, 12.628531757299762893625382596833, 13.72866932491041659903787832650, 14.130921469895109484298805895735, 14.747407059196716950491956291321, 15.6477890410474977797321379541, 16.2895291463953239389171012067, 17.17013676195825832988152694743, 18.26955375844024753964897938360, 18.85295003339888201136475688691, 19.46023489548427059162928971296, 20.29987266942114991092216873964, 21.57344199182555563401881080819, 22.0544886801285126339521461183

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