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Properties

Label 1-241-241.143-r0-0-0
Degree $1$
Conductor $241$
Sign $0.372 + 0.927i$
Analytic cond. $1.11919$
Root an. cond. $1.11919$
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  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s − 11-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + 15-s + 16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.809 − 0.587i)3-s + 4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + 8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s − 11-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + 15-s + 16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(241\)
Sign: $0.372 + 0.927i$
Analytic conductor: \(1.11919\)
Root analytic conductor: \(1.11919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{241} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 241,\ (0:\ ),\ 0.372 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075212671 + 0.7267959949i\)
\(L(\frac12)\) \(\approx\) \(1.075212671 + 0.7267959949i\)
\(L(1)\) \(\approx\) \(1.202900033 + 0.2475003451i\)
\(L(1)\) \(\approx\) \(1.202900033 + 0.2475003451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 - T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 - T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−26.05732122004885583296812556419, −24.756194664268291959981766767714, −23.692531558514201528502199799279, −23.13288334560514970444505442051, −22.74045970023870215709323042852, −21.30912069323722951261917317177, −20.64401683949531120229172329789, −19.99235794490031297396742146955, −18.586442573176612529001641499275, −17.10176952557082665563293160124, −16.36511559195522078552781008996, −15.68807683763415683977305512170, −14.906294972495591478295959222688, −13.23670860021948496374031684081, −12.87978581988125778896223309087, −11.51155113209171123452247079482, −10.94374670775419081312772948915, −9.94870555278373813508164812214, −8.198453735576626591071536473693, −7.06868877730290188253024103627, −5.96671249110451196134087092798, −4.77385658097427394599927792790, −4.194110795444317549405535989236, −3.03022914727437255450968468928, −0.74228574563317915112246905940, 1.894478450747587369411167514276, 3.08656156623040966476176223849, 4.41912628848923726827815696044, 5.62363983658096723590751482670, 6.4399753931945281626206432086, 7.349255898838102377970843115549, 8.52611529529942321678965739926, 10.763622765502596768014218962891, 10.988826614300804408763434364843, 12.30839771789904848657234262205, 12.71128423575185857412484744263, 13.88154117032596099198384655493, 15.213384177961349486103909493647, 15.7359305918151829941619476731, 16.7122184903320852815142367434, 18.1501582764431298182781347739, 18.96591845243389388238294117051, 19.66890983255840504427063594709, 21.29942962939232413185036633782, 21.81491879386994489404815062702, 22.89104155340301712463909106885, 23.52835170066256692284283325048, 24.012931418994631800141780343182, 25.32881531886470821639738707066, 25.97022644811973078106593322519

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