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Properties

Label 1-2535-2535.1343-r0-0-0
Degree $1$
Conductor $2535$
Sign $0.314 - 0.949i$
Analytic cond. $11.7724$
Root an. cond. $11.7724$
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  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.600 + 0.799i)7-s + (0.992 + 0.120i)8-s + (−0.0402 − 0.999i)11-s + (−0.354 − 0.935i)14-s + (−0.632 + 0.774i)16-s + (−0.600 + 0.799i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.979 + 0.200i)28-s + (−0.845 − 0.534i)29-s + (0.748 − 0.663i)31-s + (−0.316 − 0.948i)32-s + ⋯
L(s)  = 1  + (−0.534 + 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.600 + 0.799i)7-s + (0.992 + 0.120i)8-s + (−0.0402 − 0.999i)11-s + (−0.354 − 0.935i)14-s + (−0.632 + 0.774i)16-s + (−0.600 + 0.799i)17-s + (−0.5 + 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.979 + 0.200i)28-s + (−0.845 − 0.534i)29-s + (0.748 − 0.663i)31-s + (−0.316 − 0.948i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(11.7724\)
Root analytic conductor: \(11.7724\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2535,\ (0:\ ),\ 0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1884915251 - 0.1360902159i\)
\(L(\frac12)\) \(\approx\) \(0.1884915251 - 0.1360902159i\)
\(L(1)\) \(\approx\) \(0.5559499989 + 0.2279765181i\)
\(L(1)\) \(\approx\) \(0.5559499989 + 0.2279765181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.534 + 0.845i)T \)
7 \( 1 + (-0.600 + 0.799i)T \)
11 \( 1 + (-0.0402 - 0.999i)T \)
17 \( 1 + (-0.600 + 0.799i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.845 - 0.534i)T \)
31 \( 1 + (0.748 - 0.663i)T \)
37 \( 1 + (-0.316 + 0.948i)T \)
41 \( 1 + (0.692 + 0.721i)T \)
43 \( 1 + (0.316 + 0.948i)T \)
47 \( 1 + (0.822 - 0.568i)T \)
53 \( 1 + (-0.992 - 0.120i)T \)
59 \( 1 + (0.632 + 0.774i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (0.903 + 0.428i)T \)
71 \( 1 + (0.278 - 0.960i)T \)
73 \( 1 + (0.464 - 0.885i)T \)
79 \( 1 + (-0.568 - 0.822i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.160 + 0.987i)T \)
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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

−19.71145568581374465128086246340, −18.94385231031809094396328118503, −18.176490526725079404796350382078, −17.44124196752768614794974384208, −17.0822189210117776220590619896, −16.00050688533160178889232249234, −15.628912568063865213209090491248, −14.245508648304191313777632457845, −13.79961404705113942830657523134, −12.71802654945224458110089343105, −12.60034942300923195123351892004, −11.51301281148104233881438499398, −10.76491806427536309591120857310, −10.2417627171127610624744673976, −9.40514830307907449793140419303, −8.94958084444697838678364001048, −7.85581088694557275075476806419, −7.14207908916027001953057208185, −6.628780588205903425737969039469, −5.19447882815422877899350531876, −4.3193589814301514212195381182, −3.81121913025583463609618668655, −2.678639087669929610772816326791, −2.10311144542332017049752933390, −0.912074668707977703874554855267, 0.10802358142304061057817725522, 1.475720925005936017714047736053, 2.394500548540883412575859191599, 3.56080016788279786192601690154, 4.400766236410270488898397359965, 5.5294208041263138917836524199, 6.14208012828220404067925802866, 6.44653808179287010751846861391, 7.80062737542737383269795351656, 8.20319831117109815843560942265, 9.01079805431834361213879919637, 9.653134788346058324645762410222, 10.41053587843152206696222565863, 11.2182499094810276240895056028, 12.06811225687637626655616246891, 13.08529677361651152029076768369, 13.57705488848387961609321344404, 14.51221561743503051869849799404, 15.18712734394082083070930186373, 15.76031082336998974376699629490, 16.457381342634800730915877634344, 17.03534118923899654925377463793, 17.84815630240716135931646357454, 18.62002803230228442501028747741, 19.115448115823716630720709683967

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