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Properties

Label 1-751-751.88-r1-0-0
Degree $1$
Conductor $751$
Sign $-0.764 - 0.644i$
Analytic cond. $80.7061$
Root an. cond. $80.7061$
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  + (0.372 − 0.928i)2-s + (−0.740 − 0.672i)3-s + (−0.723 − 0.690i)4-s + (−0.624 + 0.781i)5-s + (−0.899 + 0.437i)6-s + (0.745 − 0.666i)7-s + (−0.910 + 0.414i)8-s + (0.0961 + 0.995i)9-s + (0.492 + 0.870i)10-s + (−0.783 + 0.621i)11-s + (0.0711 + 0.997i)12-s + (−0.788 − 0.614i)13-s + (−0.340 − 0.940i)14-s + (0.987 − 0.158i)15-s + (0.0460 + 0.998i)16-s + (0.799 + 0.601i)17-s + ⋯
L(s)  = 1  + (0.372 − 0.928i)2-s + (−0.740 − 0.672i)3-s + (−0.723 − 0.690i)4-s + (−0.624 + 0.781i)5-s + (−0.899 + 0.437i)6-s + (0.745 − 0.666i)7-s + (−0.910 + 0.414i)8-s + (0.0961 + 0.995i)9-s + (0.492 + 0.870i)10-s + (−0.783 + 0.621i)11-s + (0.0711 + 0.997i)12-s + (−0.788 − 0.614i)13-s + (−0.340 − 0.940i)14-s + (0.987 − 0.158i)15-s + (0.0460 + 0.998i)16-s + (0.799 + 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(751\)
Sign: $-0.764 - 0.644i$
Analytic conductor: \(80.7061\)
Root analytic conductor: \(80.7061\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{751} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 751,\ (1:\ ),\ -0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2785530960 - 0.7626903753i\)
\(L(\frac12)\) \(\approx\) \(0.2785530960 - 0.7626903753i\)
\(L(1)\) \(\approx\) \(0.6080812845 - 0.4123340047i\)
\(L(1)\) \(\approx\) \(0.6080812845 - 0.4123340047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad751 \( 1 \)
good2 \( 1 + (0.372 - 0.928i)T \)
3 \( 1 + (-0.740 - 0.672i)T \)
5 \( 1 + (-0.624 + 0.781i)T \)
7 \( 1 + (0.745 - 0.666i)T \)
11 \( 1 + (-0.783 + 0.621i)T \)
13 \( 1 + (-0.788 - 0.614i)T \)
17 \( 1 + (0.799 + 0.601i)T \)
19 \( 1 + (-0.941 + 0.336i)T \)
23 \( 1 + (-0.880 + 0.474i)T \)
29 \( 1 + (0.301 + 0.953i)T \)
31 \( 1 + (-0.507 + 0.861i)T \)
37 \( 1 + (0.859 - 0.510i)T \)
41 \( 1 + (0.929 + 0.368i)T \)
43 \( 1 + (-0.711 + 0.702i)T \)
47 \( 1 + (0.0460 - 0.998i)T \)
53 \( 1 + (0.968 + 0.248i)T \)
59 \( 1 + (-0.650 - 0.759i)T \)
61 \( 1 + (-0.268 - 0.963i)T \)
67 \( 1 + (0.316 - 0.948i)T \)
71 \( 1 + (-0.778 + 0.627i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.941 + 0.336i)T \)
83 \( 1 + (0.756 - 0.653i)T \)
89 \( 1 + (-0.252 + 0.967i)T \)
97 \( 1 + (-0.203 + 0.979i)T \)
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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

−22.65898066540568057203737290649, −21.756013804866537370350745729381, −21.213717509804715193812728615941, −20.57942555365934852001537701025, −19.07608385827365189101717235942, −18.29324049510275769897528894197, −17.38874504799288553427607859584, −16.59944685603081869814966487897, −16.17664660366330853196709464641, −15.25517912739103648744741401997, −14.774461639313372039247809232342, −13.63712891116468231736219173431, −12.50037170917975607715353362443, −11.96470635996438916391812233823, −11.218186484401442029442226782980, −9.86277238152707160124960322949, −8.97863277350287232770090829986, −8.19715620744057551356747994033, −7.40377439067910027164269717558, −6.04961974138876018468360245349, −5.428469830558914608618065170731, −4.60480027161617224361780002255, −4.08953533860800812435150500892, −2.621422049779641840220871094555, −0.60315919853923642764380718833, 0.31396054051540826858183071191, 1.57603619440762973332702985628, 2.46070156548729937862124444669, 3.66612004205997917545045584548, 4.67491786908933801087612724625, 5.43901125626695601335961381176, 6.56662514928330239549146429078, 7.67450517012899725387214607487, 8.099683756602819194156967784476, 9.96335247090227530830483229366, 10.618637895803549695755065726218, 11.02333043746897944268069127550, 12.18771839000730939385773855260, 12.49521002680357459758981272316, 13.54221289768029426541734985480, 14.531937942215874055767367340618, 14.99120144439516126362253590720, 16.338357338745562849014592767829, 17.48967418322622002392515748629, 18.03468613626548077070183235922, 18.652766495953709835736022262845, 19.72641905220828472524912083899, 19.996303348123232550647850573, 21.37747468580133373678628165735, 21.849428268074257536276499087393

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