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Properties

Label 1-2527-2527.1053-r0-0-0
Degree $1$
Conductor $2527$
Sign $-0.469 - 0.883i$
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  + (0.879 − 0.475i)2-s + (0.635 − 0.771i)3-s + (0.546 − 0.837i)4-s + (0.401 + 0.915i)5-s + (0.191 − 0.981i)6-s + (0.0825 − 0.996i)8-s + (−0.191 − 0.981i)9-s + (0.789 + 0.614i)10-s + (−0.754 − 0.656i)11-s + (−0.298 − 0.954i)12-s + (0.635 − 0.771i)13-s + (0.962 + 0.272i)15-s + (−0.401 − 0.915i)16-s + (0.998 − 0.0550i)17-s + (−0.635 − 0.771i)18-s + ⋯
L(s)  = 1  + (0.879 − 0.475i)2-s + (0.635 − 0.771i)3-s + (0.546 − 0.837i)4-s + (0.401 + 0.915i)5-s + (0.191 − 0.981i)6-s + (0.0825 − 0.996i)8-s + (−0.191 − 0.981i)9-s + (0.789 + 0.614i)10-s + (−0.754 − 0.656i)11-s + (−0.298 − 0.954i)12-s + (0.635 − 0.771i)13-s + (0.962 + 0.272i)15-s + (−0.401 − 0.915i)16-s + (0.998 − 0.0550i)17-s + (−0.635 − 0.771i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.469 - 0.883i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ -0.469 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.088555993 - 3.474099783i\)
\(L(\frac12)\) \(\approx\) \(2.088555993 - 3.474099783i\)
\(L(1)\) \(\approx\) \(1.903547923 - 1.320135040i\)
\(L(1)\) \(\approx\) \(1.903547923 - 1.320135040i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.879 - 0.475i)T \)
3 \( 1 + (0.635 - 0.771i)T \)
5 \( 1 + (0.401 + 0.915i)T \)
11 \( 1 + (-0.754 - 0.656i)T \)
13 \( 1 + (0.635 - 0.771i)T \)
17 \( 1 + (0.998 - 0.0550i)T \)
23 \( 1 + (0.635 + 0.771i)T \)
29 \( 1 + (0.998 - 0.0550i)T \)
31 \( 1 + (0.0275 + 0.999i)T \)
37 \( 1 + (0.191 - 0.981i)T \)
41 \( 1 + (0.716 - 0.697i)T \)
43 \( 1 + (-0.926 - 0.376i)T \)
47 \( 1 + (0.754 + 0.656i)T \)
53 \( 1 + (-0.945 + 0.324i)T \)
59 \( 1 + (-0.962 - 0.272i)T \)
61 \( 1 + (0.821 - 0.569i)T \)
67 \( 1 + (0.0825 - 0.996i)T \)
71 \( 1 + (-0.904 + 0.426i)T \)
73 \( 1 + (0.998 - 0.0550i)T \)
79 \( 1 + (-0.789 + 0.614i)T \)
83 \( 1 + (0.401 - 0.915i)T \)
89 \( 1 + (-0.998 - 0.0550i)T \)
97 \( 1 + (-0.821 + 0.569i)T \)
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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

−20.08791645020094121973156151345, −19.09312707720252852030178287619, −18.143691103302535064279389682846, −17.12369332925383534492414301614, −16.54724576048409514697470788075, −16.119338619462694413776458310099, −15.31597479643861031416955546773, −14.72154013632513782914182177026, −13.929720362809725839272454368053, −13.37627034914941066315149759655, −12.73578614460141455238243553364, −11.949915138523178076242361159413, −11.060458888739661686492846127152, −10.11662098562965580361787028430, −9.47402269722030268212802729414, −8.45611005451375393252817692520, −8.15671427722506522946631057685, −7.1570727598259037941371125860, −6.12924668282992729318702671866, −5.36250785226051197685407259810, −4.61268206759692341332104886148, −4.24736088703538980211556844186, −3.14029942742585672830160951258, −2.42500828183583483891114626576, −1.44983751079346330884523269629, 0.85785674965402222709956459474, 1.708772779508524915646656845358, 2.76590982544578650266821370678, 3.10495594002760676928393784719, 3.76748170964863604324047627742, 5.23616667542481087279432270550, 5.82257021992715176866278862090, 6.51018371885422586677494820126, 7.35622397532460903953299002538, 7.98544972578965022462916116794, 9.0667879362466425880732175704, 9.96211755247017160373126386591, 10.69505500148638472670303910993, 11.230770722664010456114571753200, 12.2590063925054362689246345589, 12.814894652440645201759604528405, 13.61574389995915780610873236878, 14.008123278414992781838817205056, 14.60754745889291542782797836866, 15.481723731255829729907145117170, 15.92130026451828332976836372686, 17.3099046295712103664996787938, 18.095454617316989342069350902719, 18.69185348561553335897463269144, 19.185993029700157674445800159850

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