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Properties

Label 1-1375-1375.1011-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.332 + 0.942i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.0627 + 0.998i)2-s + (−0.425 + 0.904i)3-s + (−0.992 − 0.125i)4-s + (−0.876 − 0.481i)6-s + (0.809 + 0.587i)7-s + (0.187 − 0.982i)8-s + (−0.637 − 0.770i)9-s + (0.535 − 0.844i)12-s + (0.637 + 0.770i)13-s + (−0.637 + 0.770i)14-s + (0.968 + 0.248i)16-s + (0.992 − 0.125i)17-s + (0.809 − 0.587i)18-s + (0.425 + 0.904i)19-s + (−0.876 + 0.481i)21-s + ⋯
L(s)  = 1  + (−0.0627 + 0.998i)2-s + (−0.425 + 0.904i)3-s + (−0.992 − 0.125i)4-s + (−0.876 − 0.481i)6-s + (0.809 + 0.587i)7-s + (0.187 − 0.982i)8-s + (−0.637 − 0.770i)9-s + (0.535 − 0.844i)12-s + (0.637 + 0.770i)13-s + (−0.637 + 0.770i)14-s + (0.968 + 0.248i)16-s + (0.992 − 0.125i)17-s + (0.809 − 0.587i)18-s + (0.425 + 0.904i)19-s + (−0.876 + 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.332 + 0.942i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.332 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053290982 + 1.488717812i\)
\(L(\frac12)\) \(\approx\) \(1.053290982 + 1.488717812i\)
\(L(1)\) \(\approx\) \(0.6951628747 + 0.6736178630i\)
\(L(1)\) \(\approx\) \(0.6951628747 + 0.6736178630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0627 + 0.998i)T \)
3 \( 1 + (-0.425 + 0.904i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.637 + 0.770i)T \)
17 \( 1 + (0.992 - 0.125i)T \)
19 \( 1 + (0.425 + 0.904i)T \)
23 \( 1 + (-0.929 - 0.368i)T \)
29 \( 1 + (-0.728 - 0.684i)T \)
31 \( 1 + (-0.992 + 0.125i)T \)
37 \( 1 + (0.968 + 0.248i)T \)
41 \( 1 + (0.929 - 0.368i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.187 - 0.982i)T \)
53 \( 1 + (0.876 - 0.481i)T \)
59 \( 1 + (0.535 - 0.844i)T \)
61 \( 1 + (0.929 + 0.368i)T \)
67 \( 1 + (0.728 - 0.684i)T \)
71 \( 1 + (-0.187 - 0.982i)T \)
73 \( 1 + (-0.535 - 0.844i)T \)
79 \( 1 + (0.425 - 0.904i)T \)
83 \( 1 + (0.425 + 0.904i)T \)
89 \( 1 + (0.535 + 0.844i)T \)
97 \( 1 + (0.728 + 0.684i)T \)
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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

−20.0999009402017437596454934937, −19.93927804156088304506181064952, −18.83745664660685453651940560553, −18.091341504206996001099577156948, −17.80340991545713347790990738872, −16.95652626768766519564600988701, −16.12951433604999034347433048558, −14.61422897440720282760003343962, −14.184555267548368695837551178058, −13.11550457950001848425061551783, −12.91108559523039552513006864479, −11.75280217478152203669595947535, −11.26573427841381711537999777832, −10.64005105926475343605785158708, −9.71843075044839789832492496007, −8.61481788154156441492970969819, −7.836819213978605029391906156845, −7.331098461718822092761503455567, −5.8566546625242222218465212741, −5.31003829566009257070955545577, −4.24998991757864475370254585441, −3.2672327885244709523076968305, −2.25669623722448723253957011076, −1.24134675858724297494212908381, −0.76807071276679177237267394628, 0.58839136175442496301602791618, 1.9077312581391990453270536786, 3.594027942825418063640354584056, 4.12147679154554450806843352430, 5.20073349987011130559797635282, 5.66895712942149062003476255510, 6.42622995572951816486727179699, 7.64104497628606566879187969862, 8.342133041231276796670396163715, 9.159929441524666424660100408514, 9.813865851697636116925652094257, 10.69380982143688476349740567858, 11.69276634616526028539265069329, 12.266394271116304361438800446579, 13.542550680984353449584542843196, 14.46889793033608444656087665295, 14.73360944070056707217434723146, 15.67530745234828072014127184296, 16.43120000468049299094933437130, 16.7382033607881085308819634658, 17.83676039416648483928308977390, 18.32419883335545811715804984509, 19.036153405061060434038380331734, 20.41667538747453337784042360531, 21.03778129989909339617190427257

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