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Properties

Label 1-50e2-2500.11-r1-0-0
Degree $1$
Conductor $2500$
Sign $-0.759 - 0.650i$
Analytic cond. $268.662$
Root an. cond. $268.662$
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  + (−0.656 + 0.754i)3-s + (0.187 + 0.982i)7-s + (−0.137 − 0.990i)9-s + (−0.577 − 0.816i)11-s + (−0.470 + 0.882i)13-s + (−0.332 − 0.942i)17-s + (−0.920 + 0.391i)19-s + (−0.863 − 0.503i)21-s + (0.379 − 0.925i)23-s + (0.837 + 0.546i)27-s + (0.162 + 0.986i)29-s + (0.332 + 0.942i)31-s + (0.994 + 0.100i)33-s + (0.260 + 0.965i)37-s + (−0.356 − 0.934i)39-s + ⋯
L(s)  = 1  + (−0.656 + 0.754i)3-s + (0.187 + 0.982i)7-s + (−0.137 − 0.990i)9-s + (−0.577 − 0.816i)11-s + (−0.470 + 0.882i)13-s + (−0.332 − 0.942i)17-s + (−0.920 + 0.391i)19-s + (−0.863 − 0.503i)21-s + (0.379 − 0.925i)23-s + (0.837 + 0.546i)27-s + (0.162 + 0.986i)29-s + (0.332 + 0.942i)31-s + (0.994 + 0.100i)33-s + (0.260 + 0.965i)37-s + (−0.356 − 0.934i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.759 - 0.650i$
Analytic conductor: \(268.662\)
Root analytic conductor: \(268.662\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2500,\ (1:\ ),\ -0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1563054249 + 0.4224578267i\)
\(L(\frac12)\) \(\approx\) \(-0.1563054249 + 0.4224578267i\)
\(L(1)\) \(\approx\) \(0.6885648761 + 0.2809729922i\)
\(L(1)\) \(\approx\) \(0.6885648761 + 0.2809729922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.656 + 0.754i)T \)
7 \( 1 + (0.187 + 0.982i)T \)
11 \( 1 + (-0.577 - 0.816i)T \)
13 \( 1 + (-0.470 + 0.882i)T \)
17 \( 1 + (-0.332 - 0.942i)T \)
19 \( 1 + (-0.920 + 0.391i)T \)
23 \( 1 + (0.379 - 0.925i)T \)
29 \( 1 + (0.162 + 0.986i)T \)
31 \( 1 + (0.332 + 0.942i)T \)
37 \( 1 + (0.260 + 0.965i)T \)
41 \( 1 + (0.762 - 0.647i)T \)
43 \( 1 + (0.637 + 0.770i)T \)
47 \( 1 + (0.556 - 0.830i)T \)
53 \( 1 + (-0.745 + 0.666i)T \)
59 \( 1 + (0.910 + 0.414i)T \)
61 \( 1 + (0.762 + 0.647i)T \)
67 \( 1 + (0.711 - 0.702i)T \)
71 \( 1 + (-0.938 + 0.344i)T \)
73 \( 1 + (0.979 + 0.199i)T \)
79 \( 1 + (-0.656 + 0.754i)T \)
83 \( 1 + (0.514 - 0.857i)T \)
89 \( 1 + (0.112 + 0.993i)T \)
97 \( 1 + (0.448 - 0.893i)T \)
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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

−18.9993520628909385287648184891, −17.694729693346015587426270110, −17.549355122580492175744184583037, −17.10133987876000106155660624084, −16.0464945069553157040758575386, −15.28036165227165815001791221706, −14.5567317753329882002348492201, −13.58463893061175122201806492343, −12.87769905373496440462297761357, −12.698302461237580580002268639504, −11.524805719857814507087775472874, −10.8915177131221555399278414475, −10.320704059869941540247394113066, −9.54478092030782840092861909214, −8.19411179876242705560840895551, −7.73189734307711630065945190197, −7.105386223107205571202219056825, −6.2708750581357305974767399635, −5.49033934202139900323098644164, −4.63240824817651746342913373500, −3.95538417277960369867621681703, −2.57653331079031694096712486744, −1.92255784751775514741212772177, −0.8178925893932889414538749410, −0.1122775858106457934578802829, 1.00411199001966643928378385532, 2.36968562284683086605121396744, 2.98975832501140220428950496815, 4.143978766106148558767218355027, 4.88956098475855917641202809775, 5.445686870331896218703704330687, 6.33333789856250989224903892397, 6.935584605024949425020397777475, 8.2498971735951850999195658457, 8.88337013885306623043910273491, 9.4452772350310372356146402763, 10.47427104759343524292856664531, 10.96541976566039879872113429345, 11.79789489099885294823798202600, 12.28269004316301030416667074096, 13.138621442694613210859722912211, 14.315863941803432563423835731584, 14.66205674709035890042194907324, 15.697451979812322188752778998006, 16.07210699234574419872282642922, 16.76250290695397933988187006630, 17.51454186836926456948571843329, 18.40320377327518725815304322077, 18.74867300299060029206815622867, 19.682560589258803693182385727019

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