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Properties

Label 1-2527-2527.263-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.0642 - 0.997i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (−0.227 − 0.973i)2-s + (−0.999 − 0.0183i)3-s + (−0.896 + 0.443i)4-s + (−0.991 − 0.128i)5-s + (0.209 + 0.977i)6-s + (0.635 + 0.771i)8-s + (0.999 + 0.0367i)9-s + (0.100 + 0.994i)10-s + (−0.926 + 0.376i)11-s + (0.904 − 0.426i)12-s + (0.933 + 0.359i)13-s + (0.989 + 0.146i)15-s + (0.606 − 0.794i)16-s + (−0.896 − 0.443i)17-s + (−0.191 − 0.981i)18-s + ⋯
L(s)  = 1  + (−0.227 − 0.973i)2-s + (−0.999 − 0.0183i)3-s + (−0.896 + 0.443i)4-s + (−0.991 − 0.128i)5-s + (0.209 + 0.977i)6-s + (0.635 + 0.771i)8-s + (0.999 + 0.0367i)9-s + (0.100 + 0.994i)10-s + (−0.926 + 0.376i)11-s + (0.904 − 0.426i)12-s + (0.933 + 0.359i)13-s + (0.989 + 0.146i)15-s + (0.606 − 0.794i)16-s + (−0.896 − 0.443i)17-s + (−0.191 − 0.981i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0642 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0642 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.0642 - 0.997i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.0642 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3171360498 - 0.2973843834i\)
\(L(\frac12)\) \(\approx\) \(0.3171360498 - 0.2973843834i\)
\(L(1)\) \(\approx\) \(0.4373186789 - 0.2024580872i\)
\(L(1)\) \(\approx\) \(0.4373186789 - 0.2024580872i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.227 - 0.973i)T \)
3 \( 1 + (-0.999 - 0.0183i)T \)
5 \( 1 + (-0.991 - 0.128i)T \)
11 \( 1 + (-0.926 + 0.376i)T \)
13 \( 1 + (0.933 + 0.359i)T \)
17 \( 1 + (-0.896 - 0.443i)T \)
23 \( 1 + (0.484 - 0.875i)T \)
29 \( 1 + (-0.971 + 0.236i)T \)
31 \( 1 + (0.546 - 0.837i)T \)
37 \( 1 + (-0.926 + 0.376i)T \)
41 \( 1 + (-0.979 + 0.200i)T \)
43 \( 1 + (0.100 - 0.994i)T \)
47 \( 1 + (0.209 + 0.977i)T \)
53 \( 1 + (0.209 + 0.977i)T \)
59 \( 1 + (-0.979 + 0.200i)T \)
61 \( 1 + (-0.333 + 0.942i)T \)
67 \( 1 + (0.870 - 0.492i)T \)
71 \( 1 + (0.983 + 0.182i)T \)
73 \( 1 + (0.0642 + 0.997i)T \)
79 \( 1 + (-0.912 + 0.410i)T \)
83 \( 1 + (-0.298 - 0.954i)T \)
89 \( 1 + (0.0642 - 0.997i)T \)
97 \( 1 + (-0.00918 - 0.999i)T \)
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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

−19.22560664352846330648198343603, −18.71714169157864104154003272579, −18.05051883004152263369843182716, −17.47289362324051874661148134647, −16.635056127794296805838132998772, −16.007367631513966473296812338830, −15.43493398743825095771922653238, −15.17038301043439667192448342373, −13.83336882055111452869871980598, −13.16090972401542159123789390937, −12.57266574788641175323361566934, −11.495614779590107098353028038703, −10.85621771324529768341171670702, −10.380153590092267560152179220858, −9.279825083201067870599134298332, −8.37505598356457702238850839218, −7.85304886728830827611863514371, −6.96283562326724002963588676881, −6.44380339601733432726687628672, −5.46037104902378066089765229349, −5.00587332951854003221484801969, −4.01050510370847619361802607820, −3.36507262682168771852037226789, −1.62415537868400739094783228002, −0.51107881328426826523698338105, 0.38915412200358799775660990224, 1.40226004207282160504165891657, 2.43884547359729477930764621334, 3.474630108154952032640686761238, 4.35958113785270892149319779978, 4.763039923521764143206292843623, 5.71139404769055549625917706287, 6.86666782412969958184905086003, 7.52718306482569923274260018854, 8.43590819122180490979541783346, 9.098640787097779263528467955937, 10.13917752633447917657272977058, 10.80622396089411502855532195449, 11.27460218597237768457914305098, 11.895275878125038006273706509895, 12.665866579726159797155423816074, 13.13930336429517036577392501365, 13.97565282939236131396165818274, 15.35332967825835737144645270115, 15.660108989715893068676613525006, 16.66483754484402220916251958748, 17.12523142845901595039632181160, 18.137862311341370663538496943856, 18.64083820129010983667797060113, 18.95928686115505913530743101682

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