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Properties

Label 1-315-315.122-r1-0-0
Degree $1$
Conductor $315$
Sign $0.945 + 0.325i$
Analytic cond. $33.8514$
Root an. cond. $33.8514$
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  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s − 11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (−0.5 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)32-s + (−0.5 + 0.866i)34-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + i·8-s − 11-s + (0.866 − 0.5i)13-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.866 − 0.5i)22-s + i·23-s + (−0.5 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (0.866 + 0.5i)32-s + (−0.5 + 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.945 + 0.325i$
Analytic conductor: \(33.8514\)
Root analytic conductor: \(33.8514\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (1:\ ),\ 0.945 + 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134892025 + 0.1897095034i\)
\(L(\frac12)\) \(\approx\) \(1.134892025 + 0.1897095034i\)
\(L(1)\) \(\approx\) \(0.7445181363 + 0.1190345451i\)
\(L(1)\) \(\approx\) \(0.7445181363 + 0.1190345451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 - T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−25.22779958800741574259556618835, −24.083443160746168675923791877164, −23.19758852666365627687416989088, −21.97366484891788472767121522412, −21.03965067699119193151858505492, −20.59596919029998684825711287169, −19.300047754556320317541803234524, −18.740449804593748649732198293700, −17.84684704642106498533294381851, −16.90580371806362078141063334887, −16.038986714542612846071871687949, −15.18016397498711570742618578357, −13.70348095469963104060639494150, −12.80837833035783695601122291757, −11.852202222613577747580640653064, −10.78387065171288972675910151966, −10.184426888453444082680032243413, −8.9166651944250175779970068508, −8.21266367263178679155687356363, −7.130860952603233999045547689861, −6.01502998139293893976068752422, −4.44983176148586654825324857480, −3.20097018519621495532979114956, −2.10467121408246983300180212445, −0.74376723420134199942158816018, 0.708961673733062205329981992061, 2.05886076684994699923508350502, 3.491292208128608780909642205804, 5.26815084772930221589295780524, 5.92118756239250297826265253378, 7.29390463325525364912368910720, 8.01091466838920987984049377866, 8.986839008992272303914694066819, 10.12643866310701161947838306039, 10.76604212435107463260431716817, 11.92099976712458121704357506419, 13.20093709990687690349983975464, 14.2325205032321889075036610914, 15.29743191301729185929152307651, 15.995007336999263423107482993168, 16.863341594042704075044431172203, 17.90838117021227697330562699959, 18.58109269049992323469300363026, 19.36123288138585556000246367857, 20.63247545184950709207133096880, 21.013123709827276670490207781692, 22.66966212173271424576577713349, 23.43280711101488949440332267332, 24.17481190215363937795891609, 25.41695544138065993220388714455

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