Properties

Label 1-195-195.89-r0-0-0
Degree $1$
Conductor $195$
Sign $0.533 - 0.846i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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 + (0.866 − 0.5i)7-s i·8-s + (−0.866 − 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)28-s + (0.5 − 0.866i)29-s + i·31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s i·8-s + (−0.866 − 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (0.866 + 0.5i)28-s + (0.5 − 0.866i)29-s + i·31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.533 - 0.846i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.533 - 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7224752272 - 0.3987257138i\)
\(L(\frac12)\) \(\approx\) \(0.7224752272 - 0.3987257138i\)
\(L(1)\) \(\approx\) \(0.7540055343 - 0.2329068299i\)
\(L(1)\) \(\approx\) \(0.7540055343 - 0.2329068299i\)

Euler product

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

−27.28629468634343745255906855396, −26.17694411772961992468984917008, −25.32374069455042481185552741854, −24.494872422536530771184710698423, −23.639451173308039319626587187460, −22.639206916943807892370352448725, −21.074996741589023373772287020764, −20.51979338837583386726658863670, −19.246178951777605741027479440233, −18.226044243496420608740506303539, −17.79363087098301786198009119218, −16.52610671314099830256299693214, −15.6070523536056925987483659567, −14.76782868374278147364625885831, −13.74135637452607346895435221559, −12.121545548764343682794526639300, −11.18840200019664910414565512997, −10.06733741141620636044831885939, −9.10063715511244686891820112535, −7.92769557458567971496597598363, −7.26640427092044111187235844616, −5.676186363351765547399247265495, −4.92804310702322946306392730430, −2.7484056417810336456903031424, −1.397585913064869499700104611140, 0.99628479471565851305740786538, 2.4409964922988763156805640598, 3.75823815407918079168835147559, 5.226910155547674524294639965092, 6.88534915531915462911463381531, 7.953997317310953428936264137254, 8.67664405519975396371158359949, 10.1384054636680703538197739374, 10.80901146830070208585803480338, 11.79863644943899044243645832847, 12.94113577992954095854790350586, 14.05666512534262680157119460045, 15.413741462008383498514157865434, 16.451917306658588509295487776686, 17.39505540265263322954696828403, 18.19409029139475771203514522034, 19.134641662308088646056679814300, 20.15391282649519742021646406816, 21.03894304135804381411490574480, 21.65618648374748390339854728803, 23.10674824216080661042721725524, 24.19070413233920145211080566622, 25.03933267037368793079441466987, 26.46536907875145392018231713367, 26.615112416453629269982743361485

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