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Properties

Label 1-55e2-3025.804-r0-0-0
Degree $1$
Conductor $3025$
Sign $-0.571 - 0.820i$
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  + (0.466 − 0.884i)2-s + (−0.309 − 0.951i)3-s + (−0.564 − 0.825i)4-s + (−0.985 − 0.170i)6-s + (−0.841 − 0.540i)7-s + (−0.993 + 0.113i)8-s + (−0.809 + 0.587i)9-s + (−0.610 + 0.791i)12-s + (−0.610 + 0.791i)13-s + (−0.870 + 0.491i)14-s + (−0.362 + 0.931i)16-s + (0.921 − 0.389i)17-s + (0.142 + 0.989i)18-s + (0.516 + 0.856i)19-s + (−0.254 + 0.967i)21-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)2-s + (−0.309 − 0.951i)3-s + (−0.564 − 0.825i)4-s + (−0.985 − 0.170i)6-s + (−0.841 − 0.540i)7-s + (−0.993 + 0.113i)8-s + (−0.809 + 0.587i)9-s + (−0.610 + 0.791i)12-s + (−0.610 + 0.791i)13-s + (−0.870 + 0.491i)14-s + (−0.362 + 0.931i)16-s + (0.921 − 0.389i)17-s + (0.142 + 0.989i)18-s + (0.516 + 0.856i)19-s + (−0.254 + 0.967i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (804, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ -0.571 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5806633158 - 1.111643607i\)
\(L(\frac12)\) \(\approx\) \(0.5806633158 - 1.111643607i\)
\(L(1)\) \(\approx\) \(0.6740838996 - 0.6866624448i\)
\(L(1)\) \(\approx\) \(0.6740838996 - 0.6866624448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.466 - 0.884i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.610 + 0.791i)T \)
17 \( 1 + (0.921 - 0.389i)T \)
19 \( 1 + (0.516 + 0.856i)T \)
23 \( 1 + (0.998 + 0.0570i)T \)
29 \( 1 + (-0.921 - 0.389i)T \)
31 \( 1 + (-0.564 + 0.825i)T \)
37 \( 1 + (-0.941 + 0.336i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.985 - 0.170i)T \)
53 \( 1 + (0.254 - 0.967i)T \)
59 \( 1 + (-0.466 - 0.884i)T \)
61 \( 1 + (0.696 - 0.717i)T \)
67 \( 1 + (0.985 + 0.170i)T \)
71 \( 1 + (-0.921 - 0.389i)T \)
73 \( 1 + (0.998 + 0.0570i)T \)
79 \( 1 + (0.993 + 0.113i)T \)
83 \( 1 + (-0.774 + 0.633i)T \)
89 \( 1 + (0.0855 + 0.996i)T \)
97 \( 1 + (0.736 - 0.676i)T \)
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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

−19.231801476462278942664529032008, −18.41942728916173685976184024562, −17.58677678891683174029724213716, −16.96447327522576778260339792340, −16.44344367788834147455972539719, −15.709460141193220277609549840779, −15.15821149156720179215939248591, −14.740449271441601280147402963743, −13.83016416649407609637903354722, −12.88369355312597478015674546739, −12.453040975960532346354052354495, −11.637627196771373656220348694, −10.7265823595311265451656278663, −9.8407489341937706656886460539, −9.22492357523170366693267119117, −8.70780332941264023846626801431, −7.58733225129717774335188033234, −6.99009064965576778277697470572, −5.86190227798221704078976898356, −5.58501694136552807308982111202, −4.86685000357867133870030076579, −3.89406262990551869009012606461, −3.20905544248159991676011072724, −2.61583232528356288594371744811, −0.58908261678117671172379910091, 0.6188067625900831950590396141, 1.47645793443562632815122751488, 2.29431414005273450610990023551, 3.23924887523234098531893113506, 3.797103590150267109002416625014, 5.05221109250777656711649623664, 5.50688973014737399572598886780, 6.523155578353497026399347115655, 7.061006887208180478405615005341, 7.90507518040900089863149710240, 9.02486324360920749918759254984, 9.66792516785308291243277219144, 10.39502490198005740916201910501, 11.19767037000201003802460905742, 11.88762166999250678358490485065, 12.49452888932642000682269865249, 12.96855146279891032520390272455, 13.90985026063307369334881063850, 14.12057315940236878215012497838, 15.044217236690725280430162299889, 16.204844105152217893518431099526, 16.82222698319898702517780721619, 17.48112306970396300406301708929, 18.526068831549387823009070101679, 18.88174012719052086321545510454

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