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Properties

Label 1-751-751.486-r1-0-0
Degree $1$
Conductor $751$
Sign $0.884 + 0.466i$
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  + (−0.995 − 0.0920i)2-s + (−0.926 − 0.375i)3-s + (0.983 + 0.183i)4-s + (0.630 + 0.775i)5-s + (0.888 + 0.459i)6-s + (−0.899 + 0.437i)7-s + (−0.962 − 0.272i)8-s + (0.717 + 0.696i)9-s + (−0.556 − 0.830i)10-s + (−0.604 + 0.796i)11-s + (−0.842 − 0.539i)12-s + (0.549 + 0.835i)13-s + (0.935 − 0.352i)14-s + (−0.293 − 0.956i)15-s + (0.932 + 0.360i)16-s + (−0.433 − 0.901i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0920i)2-s + (−0.926 − 0.375i)3-s + (0.983 + 0.183i)4-s + (0.630 + 0.775i)5-s + (0.888 + 0.459i)6-s + (−0.899 + 0.437i)7-s + (−0.962 − 0.272i)8-s + (0.717 + 0.696i)9-s + (−0.556 − 0.830i)10-s + (−0.604 + 0.796i)11-s + (−0.842 − 0.539i)12-s + (0.549 + 0.835i)13-s + (0.935 − 0.352i)14-s + (−0.293 − 0.956i)15-s + (0.932 + 0.360i)16-s + (−0.433 − 0.901i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(751\)
Sign: $0.884 + 0.466i$
Analytic conductor: \(80.7061\)
Root analytic conductor: \(80.7061\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{751} (486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 751,\ (1:\ ),\ 0.884 + 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6778313664 + 0.1678080652i\)
\(L(\frac12)\) \(\approx\) \(0.6778313664 + 0.1678080652i\)
\(L(1)\) \(\approx\) \(0.5150167668 + 0.05169804306i\)
\(L(1)\) \(\approx\) \(0.5150167668 + 0.05169804306i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad751 \( 1 \)
good2 \( 1 + (-0.995 - 0.0920i)T \)
3 \( 1 + (-0.926 - 0.375i)T \)
5 \( 1 + (0.630 + 0.775i)T \)
7 \( 1 + (-0.899 + 0.437i)T \)
11 \( 1 + (-0.604 + 0.796i)T \)
13 \( 1 + (0.549 + 0.835i)T \)
17 \( 1 + (-0.433 - 0.901i)T \)
19 \( 1 + (-0.923 + 0.383i)T \)
23 \( 1 + (-0.687 - 0.726i)T \)
29 \( 1 + (0.767 - 0.640i)T \)
31 \( 1 + (0.440 - 0.897i)T \)
37 \( 1 + (-0.410 - 0.911i)T \)
41 \( 1 + (0.992 + 0.125i)T \)
43 \( 1 + (0.998 - 0.0502i)T \)
47 \( 1 + (0.932 - 0.360i)T \)
53 \( 1 + (-0.425 - 0.904i)T \)
59 \( 1 + (0.813 - 0.580i)T \)
61 \( 1 + (-0.570 + 0.821i)T \)
67 \( 1 + (0.846 + 0.532i)T \)
71 \( 1 + (0.656 - 0.754i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.923 + 0.383i)T \)
83 \( 1 + (-0.832 + 0.553i)T \)
89 \( 1 + (-0.455 - 0.890i)T \)
97 \( 1 + (-0.0711 - 0.997i)T \)
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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

−21.88821518622403196548326395302, −21.40214704506937185656945337126, −20.51805709656023323745039229538, −19.71469884141575305790646555491, −18.85912396677352490850060261562, −17.77475113957318576505416820029, −17.39129712800150691738231478588, −16.61611929372712473046424052743, −15.82760508357074939670669677269, −15.53297672338091668427443592424, −13.86094635113685389680982634815, −12.826522358703291314305901169078, −12.3286104610553082997473271834, −10.86352926054970999560461839719, −10.59005317020265469167221736006, −9.74820499363034687203205612331, −8.85899094208332799325069504952, −8.06185915360215747234596440129, −6.69004664830565377628169687778, −6.0657231221360004801478068631, −5.40172154689138346097442980005, −4.02890343499932238446773169458, −2.805562097081147289622115609599, −1.293616036868909572393679225913, −0.51422322977251849136120456781, 0.522826405752120138226560844596, 2.14350489340447170182798568501, 2.42685699889363327058141823009, 4.11499903644032581164457829421, 5.65555584972738314687851558810, 6.41648117630514239529347993760, 6.85156272659609378910295917360, 7.84905070819527517428657098757, 9.14018188822114318917199964410, 9.917488506362988899326333167, 10.54418244456053376428324593515, 11.36380996758587685706342040694, 12.25523090067547834249226299968, 12.97660344105526403924391156620, 14.064149824889015761817856049110, 15.378956060683054086089510078883, 15.984473944258247992730256980548, 16.775629741440545303548993915949, 17.691171507830329824158290378207, 18.21747203803459260996504756526, 18.85826848167156769904057917795, 19.44080966220931998496340142477, 20.82333697409578985321795504817, 21.38756141119671635418981795559, 22.441404572644708037085923512623

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