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Properties

Label 1-55e2-3025.48-r1-0-0
Degree $1$
Conductor $3025$
Sign $-0.442 - 0.896i$
Analytic cond. $325.081$
Root an. cond. $325.081$
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  + (−0.755 + 0.654i)2-s + (−0.951 + 0.309i)3-s + (0.142 − 0.989i)4-s + (0.516 − 0.856i)6-s + (−0.791 − 0.610i)7-s + (0.540 + 0.841i)8-s + (0.809 − 0.587i)9-s + (0.170 + 0.985i)12-s + (0.884 − 0.466i)13-s + (0.998 − 0.0570i)14-s + (−0.959 − 0.281i)16-s + (0.676 + 0.736i)17-s + (−0.226 + 0.974i)18-s + (−0.415 + 0.909i)19-s + (0.941 + 0.336i)21-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (−0.951 + 0.309i)3-s + (0.142 − 0.989i)4-s + (0.516 − 0.856i)6-s + (−0.791 − 0.610i)7-s + (0.540 + 0.841i)8-s + (0.809 − 0.587i)9-s + (0.170 + 0.985i)12-s + (0.884 − 0.466i)13-s + (0.998 − 0.0570i)14-s + (−0.959 − 0.281i)16-s + (0.676 + 0.736i)17-s + (−0.226 + 0.974i)18-s + (−0.415 + 0.909i)19-s + (0.941 + 0.336i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(325.081\)
Root analytic conductor: \(325.081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (1:\ ),\ -0.442 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07718803587 + 0.1241793803i\)
\(L(\frac12)\) \(\approx\) \(-0.07718803587 + 0.1241793803i\)
\(L(1)\) \(\approx\) \(0.4439107731 + 0.1813008141i\)
\(L(1)\) \(\approx\) \(0.4439107731 + 0.1813008141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.755 + 0.654i)T \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (-0.791 - 0.610i)T \)
13 \( 1 + (0.884 - 0.466i)T \)
17 \( 1 + (0.676 + 0.736i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (-0.825 - 0.564i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.466 + 0.884i)T \)
37 \( 1 + (0.884 + 0.466i)T \)
41 \( 1 + (-0.921 + 0.389i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (-0.389 + 0.921i)T \)
53 \( 1 + (-0.999 - 0.0285i)T \)
59 \( 1 + (-0.974 + 0.226i)T \)
61 \( 1 + (0.974 - 0.226i)T \)
67 \( 1 + (0.226 - 0.974i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (0.999 - 0.0285i)T \)
79 \( 1 + (-0.774 + 0.633i)T \)
83 \( 1 + (0.825 - 0.564i)T \)
89 \( 1 + (-0.993 + 0.113i)T \)
97 \( 1 + (-0.967 + 0.254i)T \)
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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

−18.48587539863336020717648935852, −17.88543867714240009732577817070, −17.06028729900491063912026002417, −16.48981288506200782088798346777, −15.8812845700975412622887045264, −15.306880873299616784048024889784, −13.77272529506313142660162646944, −13.25139951716138168063300131568, −12.56698279246030197572662335390, −11.816642395684357076454806736, −11.415500386938069840983789070267, −10.686821361963887071822936404635, −9.74418209996618532096173253724, −9.395614363638204649901602564582, −8.42480169480544304970739322568, −7.58429348742053352211698475172, −6.843056916985428559640136704399, −6.14107228274584713351634460510, −5.390629398117012195369059238000, −4.23929502240887965293039715612, −3.53168512672588870471867814463, −2.436534734901775874652644428297, −1.77937233479927135097015444840, −0.64363554038284343996829715985, −0.05921642802049047316102446470, 1.01101467563018178888449804072, 1.59869071176945958976260914413, 3.230109754359945150170973469391, 4.01610393616856353125357789941, 4.92538856869556383725687607335, 5.94805874856438626311112609411, 6.175089509275762155498615156676, 6.94951587791043990530703564661, 7.86029804170822860237954736616, 8.50313311781747818339031580619, 9.63414488493931633222473816270, 9.99698269165226033998577096060, 10.77833352201360414794380939171, 11.118284539467094328376454656598, 12.42594344101288634176547834587, 12.812136032469739167200035507893, 13.91513609272760492147235803654, 14.647466064292432624029703201463, 15.421811508116809066387224107825, 16.28032725571532412995315113118, 16.409324905807524250878836971899, 17.126997112008051808757063786038, 17.88218384347623983103658799040, 18.50088979270318271280421433232, 19.04877055250682456484614791728

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