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Properties

Label 1-1205-1205.1142-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.626 + 0.779i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.5i)4-s − i·6-s + (−0.130 + 0.991i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.991 − 0.130i)11-s + (0.965 − 0.258i)12-s + (−0.130 − 0.991i)13-s + (−0.991 + 0.130i)14-s + (0.5 − 0.866i)16-s + (−0.923 − 0.382i)17-s + (−0.258 + 0.965i)18-s + (−0.130 + 0.991i)19-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.5i)4-s − i·6-s + (−0.130 + 0.991i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.5i)9-s + (−0.991 − 0.130i)11-s + (0.965 − 0.258i)12-s + (−0.130 − 0.991i)13-s + (−0.991 + 0.130i)14-s + (0.5 − 0.866i)16-s + (−0.923 − 0.382i)17-s + (−0.258 + 0.965i)18-s + (−0.130 + 0.991i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.626 + 0.779i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6829268092 + 0.3273384835i\)
\(L(\frac12)\) \(\approx\) \(0.6829268092 + 0.3273384835i\)
\(L(1)\) \(\approx\) \(0.6366174642 + 0.3162110239i\)
\(L(1)\) \(\approx\) \(0.6366174642 + 0.3162110239i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 + (-0.130 + 0.991i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (-0.130 - 0.991i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (-0.130 + 0.991i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (-0.258 - 0.965i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.608 - 0.793i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−20.928923262928442295001500756295, −20.629810330152776267948032009582, −19.485287513872536345441212565115, −18.9289222322026557974926606298, −17.90799392798474048368688719352, −17.41192158643392022996293146482, −16.625960584623492347687126038535, −15.64147967791167018483342282567, −14.89092810895298247846627994672, −13.70204236116475092754676623834, −13.16067976431301321956578628701, −12.50802112143893083738353242136, −11.25510763053536266439939984858, −11.15858043435472358218724277053, −10.239419687050091493971054452173, −9.5791825241244920935083925797, −8.63925387196660539498657455417, −7.246530697006241599513538614270, −6.55996485149235292960011066767, −5.40860809393108600110789672902, −4.61658536389702494033202915254, −4.121328294857696104461501738472, −2.96132641233644654810394420123, −1.79355847499338986083917382730, −0.68918518535468286372986741924, 0.51936141160675408357343859673, 2.25534536141787715317289909022, 3.31994300413766648883816322323, 4.735487041824056439454925612729, 5.1928842630452424868938981365, 6.02187506917438371327894221936, 6.59099149104083932939714038670, 7.731438186722090692342820733245, 8.229741888920328030614874515317, 9.35445350012476369251796645626, 10.23438353161480744564261667620, 11.19671773580125359657964095722, 12.15129031253495761370146931155, 12.91755392435356340523294739314, 13.21060615737110288240832318799, 14.51794135441569465345298714485, 15.35942788471178096144596573964, 15.87611100970031321787893190680, 16.52242506450444638966166595531, 17.52929448009078264254139756179, 18.0394817491070682819138874283, 18.590818804886124454711850298857, 19.428772143623149886289864230152, 20.98161193014906515597533282906, 21.409346052479955137119346147

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