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Properties

Label 1-55e2-3025.1906-r0-0-0
Degree $1$
Conductor $3025$
Sign $0.999 + 0.00207i$
Analytic cond. $14.0480$
Root an. cond. $14.0480$
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  + (−0.736 + 0.676i)2-s + (0.309 + 0.951i)3-s + (0.0855 − 0.996i)4-s + (−0.870 − 0.491i)6-s + (0.696 − 0.717i)7-s + (0.610 + 0.791i)8-s + (−0.809 + 0.587i)9-s + (0.974 − 0.226i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)14-s + (−0.985 − 0.170i)16-s + (−0.362 + 0.931i)17-s + (0.198 − 0.980i)18-s + (−0.362 − 0.931i)19-s + (0.897 + 0.441i)21-s + ⋯
L(s)  = 1  + (−0.736 + 0.676i)2-s + (0.309 + 0.951i)3-s + (0.0855 − 0.996i)4-s + (−0.870 − 0.491i)6-s + (0.696 − 0.717i)7-s + (0.610 + 0.791i)8-s + (−0.809 + 0.587i)9-s + (0.974 − 0.226i)12-s + (0.0855 − 0.996i)13-s + (−0.0285 + 0.999i)14-s + (−0.985 − 0.170i)16-s + (−0.362 + 0.931i)17-s + (0.198 − 0.980i)18-s + (−0.362 − 0.931i)19-s + (0.897 + 0.441i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00207i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1906, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ 0.999 + 0.00207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054808486 + 0.001095464390i\)
\(L(\frac12)\) \(\approx\) \(1.054808486 + 0.001095464390i\)
\(L(1)\) \(\approx\) \(0.7715437433 + 0.2834274498i\)
\(L(1)\) \(\approx\) \(0.7715437433 + 0.2834274498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.736 + 0.676i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (0.696 - 0.717i)T \)
13 \( 1 + (0.0855 - 0.996i)T \)
17 \( 1 + (-0.362 + 0.931i)T \)
19 \( 1 + (-0.362 - 0.931i)T \)
23 \( 1 + (0.696 + 0.717i)T \)
29 \( 1 + (-0.998 + 0.0570i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.516 - 0.856i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.993 + 0.113i)T \)
53 \( 1 + (0.696 - 0.717i)T \)
59 \( 1 + (0.993 + 0.113i)T \)
61 \( 1 + (-0.870 + 0.491i)T \)
67 \( 1 + (-0.736 + 0.676i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.941 + 0.336i)T \)
83 \( 1 + (-0.466 - 0.884i)T \)
89 \( 1 + (-0.362 + 0.931i)T \)
97 \( 1 + (0.610 + 0.791i)T \)
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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

−18.80691643935500745357971240977, −18.471964996487389501238204961519, −18.07120971564956517987549443319, −17.0136254639926377535217765841, −16.6823307531268562260758387336, −15.59058580851010215585690503894, −14.7043881886327206961448081955, −14.033844374793172678902861018530, −13.286285582302162359633659401346, −12.46275964809161245491481556083, −11.98365169190926829265941393647, −11.34783350964357759564536005908, −10.72526705294961441974716486801, −9.53122899075329290578362014246, −8.93363652483192364150808840050, −8.48108434909864779159832202862, −7.63164399380874051532840788091, −7.01987032781772368025895477467, −6.2292202485358212304415565470, −5.14934038043961404642375772537, −4.15637621427110628293884153383, −3.18881774713979624244658045207, −2.30237204229920608712286513956, −1.8572952397142574479242430463, −0.9715555374686580835983572224, 0.4458383460226268770058365992, 1.63179729267650194624051059500, 2.57357227425989622868788489336, 3.75039382734708587099147610157, 4.43782296621671153249699432517, 5.31024616512110199404143605326, 5.81604474962752641640612808469, 6.98706217258706789863926227823, 7.69305645533125068861661045069, 8.30415233612963097229198061681, 9.01101710382141125492803760151, 9.675635127729443661745005851538, 10.53544705750375026940021304771, 10.92221944076989743839793777153, 11.48126437823546939248359998308, 13.10503116057146078072771502257, 13.51416694279391007367115896480, 14.5908598570939783018246050340, 15.031645055796225833808768007948, 15.38093129795657969606449368490, 16.4256592136591886415669380804, 16.88535165031743380915790125859, 17.56638374073421641080751676351, 18.033534762444561541721527386316, 19.165993100085281392094440730266

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