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Properties

Label 1-605-605.224-r0-0-0
Degree $1$
Conductor $605$
Sign $0.318 + 0.947i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (−0.774 − 0.633i)2-s + (0.809 + 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.254 − 0.967i)6-s + (−0.0855 + 0.996i)7-s + (0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (−0.415 + 0.909i)12-s + (−0.993 − 0.113i)13-s + (0.696 − 0.717i)14-s + (−0.921 + 0.389i)16-s + (0.564 − 0.825i)17-s + (0.362 − 0.931i)18-s + (0.941 + 0.336i)19-s + (−0.654 + 0.755i)21-s + ⋯
L(s)  = 1  + (−0.774 − 0.633i)2-s + (0.809 + 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.254 − 0.967i)6-s + (−0.0855 + 0.996i)7-s + (0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (−0.415 + 0.909i)12-s + (−0.993 − 0.113i)13-s + (0.696 − 0.717i)14-s + (−0.921 + 0.389i)16-s + (0.564 − 0.825i)17-s + (0.362 − 0.931i)18-s + (0.941 + 0.336i)19-s + (−0.654 + 0.755i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.318 + 0.947i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ 0.318 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9374805910 + 0.6737096587i\)
\(L(\frac12)\) \(\approx\) \(0.9374805910 + 0.6737096587i\)
\(L(1)\) \(\approx\) \(0.9291378244 + 0.1878593590i\)
\(L(1)\) \(\approx\) \(0.9291378244 + 0.1878593590i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.774 - 0.633i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.0855 + 0.996i)T \)
13 \( 1 + (-0.993 - 0.113i)T \)
17 \( 1 + (0.564 - 0.825i)T \)
19 \( 1 + (0.941 + 0.336i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.610 - 0.791i)T \)
31 \( 1 + (-0.736 + 0.676i)T \)
37 \( 1 + (0.870 + 0.491i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.362 + 0.931i)T \)
53 \( 1 + (0.921 + 0.389i)T \)
59 \( 1 + (-0.998 + 0.0570i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (-0.974 - 0.226i)T \)
79 \( 1 + (-0.985 + 0.170i)T \)
83 \( 1 + (-0.516 - 0.856i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.897 + 0.441i)T \)
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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

−23.40786515582517368517443517319, −22.21150133908910988748960412945, −20.89340475717132650444584541308, −20.0097580131558362669242797469, −19.660639041383588414584794766228, −18.72634810199494489828472175224, −17.996081235278960333875038764228, −17.044471818757575060850613016596, −16.48618301875944744800225543849, −15.22745969426524553194819267722, −14.58800281368278269864697876172, −13.89006285693045958822929125522, −12.9829368798733826832493138889, −11.88902931971706363187708531819, −10.611173788732379924546066378542, −9.88169850274288630068199171165, −9.01623349776520962905770408103, −8.10017658793534891011866460750, −7.2509435836501901814473338046, −6.859816389514259491046011447394, −5.55533746223551186240677070422, −4.30166865497517981107361572725, −3.00714802983680851843389512264, −1.7889491640521815739956645739, −0.714063288215434928846511136386, 1.486981461294584734261638390547, 2.75571400787889305745960125978, 3.0836268586765236371751881139, 4.48747388607456981288691009684, 5.49523952328062124875184537756, 7.179535929318774596985229259802, 7.88821976279925351740883351370, 8.837487608538448930010798468388, 9.602023079780209601276344784863, 10.0248344091241328373346320107, 11.3195797256451085838351703764, 12.03597013317798803245225095846, 12.96944477131264619340619635169, 14.00452087290416998789624956380, 14.98790563800678749475667997491, 15.83146173284772340873407552314, 16.5042227769094960565299994667, 17.56963251387405942431016904700, 18.537533807501248612598088011561, 19.131826864142085955007499590135, 19.94802469965565480301776268358, 20.63251718914185461063785716120, 21.54124656875418672101517032823, 21.94513241018131138313624961982, 22.89926121994992885198402738982

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