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Properties

Label 1-6145-6145.5017-r1-0-0
Degree $1$
Conductor $6145$
Sign $-0.791 + 0.610i$
Analytic cond. $660.371$
Root an. cond. $660.371$
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 − 0.476i)2-s + (−0.679 + 0.733i)3-s + (0.546 − 0.837i)4-s + (−0.248 + 0.968i)6-s + (0.292 − 0.956i)7-s + (0.0817 − 0.996i)8-s + (−0.0766 − 0.997i)9-s + (−0.760 + 0.648i)11-s + (0.243 + 0.969i)12-s + (−0.567 + 0.823i)13-s + (−0.198 − 0.980i)14-s + (−0.402 − 0.915i)16-s + (0.716 + 0.698i)17-s + (−0.542 − 0.840i)18-s + (0.904 + 0.425i)19-s + ⋯
L(s)  = 1  + (0.879 − 0.476i)2-s + (−0.679 + 0.733i)3-s + (0.546 − 0.837i)4-s + (−0.248 + 0.968i)6-s + (0.292 − 0.956i)7-s + (0.0817 − 0.996i)8-s + (−0.0766 − 0.997i)9-s + (−0.760 + 0.648i)11-s + (0.243 + 0.969i)12-s + (−0.567 + 0.823i)13-s + (−0.198 − 0.980i)14-s + (−0.402 − 0.915i)16-s + (0.716 + 0.698i)17-s + (−0.542 − 0.840i)18-s + (0.904 + 0.425i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.791 + 0.610i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (5017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ -0.791 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1443799033 + 0.4236420468i\)
\(L(\frac12)\) \(\approx\) \(0.1443799033 + 0.4236420468i\)
\(L(1)\) \(\approx\) \(1.271668361 - 0.1772855529i\)
\(L(1)\) \(\approx\) \(1.271668361 - 0.1772855529i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (0.879 - 0.476i)T \)
3 \( 1 + (-0.679 + 0.733i)T \)
7 \( 1 + (0.292 - 0.956i)T \)
11 \( 1 + (-0.760 + 0.648i)T \)
13 \( 1 + (-0.567 + 0.823i)T \)
17 \( 1 + (0.716 + 0.698i)T \)
19 \( 1 + (0.904 + 0.425i)T \)
23 \( 1 + (0.416 - 0.909i)T \)
29 \( 1 + (0.950 + 0.311i)T \)
31 \( 1 + (-0.00511 + 0.999i)T \)
37 \( 1 + (-0.0409 + 0.999i)T \)
41 \( 1 + (-0.856 - 0.516i)T \)
43 \( 1 + (0.777 + 0.629i)T \)
47 \( 1 + (-0.982 - 0.188i)T \)
53 \( 1 + (-0.989 + 0.147i)T \)
59 \( 1 + (0.690 + 0.723i)T \)
61 \( 1 + (0.137 - 0.990i)T \)
67 \( 1 + (-0.843 + 0.537i)T \)
71 \( 1 + (0.393 + 0.919i)T \)
73 \( 1 + (0.524 - 0.851i)T \)
79 \( 1 + (-0.996 + 0.0817i)T \)
83 \( 1 + (-0.360 - 0.932i)T \)
89 \( 1 + (0.563 + 0.826i)T \)
97 \( 1 + (0.998 + 0.0562i)T \)
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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

−17.28142219886410949348576219447, −16.4882565387801392674832353563, −15.78094899792078887852438608962, −15.50247956835969030597521555792, −14.51641561974597213082801259202, −13.94071667639165520856337097562, −13.19924288910345424444838176394, −12.79404535182985695917378958968, −11.98488335811470818301028011533, −11.61245055974348158209644099400, −11.02835494278310603913328267701, −10.09821536605145607879623443928, −9.1268168845178171142186846569, −8.092028901287175676866210648926, −7.79798597591467402477094031780, −7.15672067256045274914458726827, −6.2538734911761454343942426540, −5.54696438065891775467642576232, −5.293807789783873636551522586538, −4.70292367354627159508823030769, −3.316324972002665038607684634561, −2.787471853558920496493052599783, −2.14324471064735618219080246289, −1.01455769350459966133744624437, −0.0497399400991161552031696391, 1.03692544973422576687507815501, 1.63302270596014134909524265333, 2.82785887211710935214872056705, 3.46566271494045302336053562417, 4.26087018939708669631347616817, 4.875304880240597524095512857293, 5.149889874531097473316847604293, 6.21118708862256329737333278786, 6.81907738182342931412372488106, 7.454187329100941028150670479071, 8.47532428899394547353215447219, 9.61679417008090758203576431347, 10.111879657466730271561586327579, 10.48456954074401020143166601227, 11.16901833684833620872610489518, 11.947734647744092737046547337883, 12.35905300541959073090533054945, 13.072185979769623003687547071, 13.98225154876824159641218373506, 14.50797573530595796865633100897, 14.94227445165152571296066361925, 15.936150585577197445143837543935, 16.24789754778310248299274270599, 17.01571962661603130653634776019, 17.64275144843218728493961094662

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