Properties

Label 1-195-195.47-r1-0-0
Degree $1$
Conductor $195$
Sign $0.966 + 0.256i$
Analytic cond. $20.9556$
Root an. cond. $20.9556$
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  + 2-s + 4-s + 7-s + 8-s + i·11-s + 14-s + 16-s + i·17-s i·19-s + i·22-s i·23-s + 28-s + 29-s + i·31-s + 32-s + ⋯
L(s)  = 1  + 2-s + 4-s + 7-s + 8-s + i·11-s + 14-s + 16-s + i·17-s i·19-s + i·22-s i·23-s + 28-s + 29-s + i·31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(20.9556\)
Root analytic conductor: \(20.9556\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (1:\ ),\ 0.966 + 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.154272172 + 0.5422164511i\)
\(L(\frac12)\) \(\approx\) \(4.154272172 + 0.5422164511i\)
\(L(1)\) \(\approx\) \(2.289861393 + 0.1425616142i\)
\(L(1)\) \(\approx\) \(2.289861393 + 0.1425616142i\)

Euler product

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

−26.80702397246784271226343071250, −25.40961891630136022353309475267, −24.66095937599771657853098812616, −23.84894709901981558547809693122, −23.01405465355358378317281294363, −21.88969845060181793744097262835, −21.15429268123496804920893025819, −20.377897487302434547524306721364, −19.22778338730797082738118550270, −18.07708238554148652674536645283, −16.798026016040682380811650724805, −15.94619830268272538436705109334, −14.804461086087069551907010302415, −14.02734003634959196503183300958, −13.19365120155182791724089662268, −11.75617549227333616075955358397, −11.33099805929893682600447298340, −10.03081356951816795898394259575, −8.36192668101016826764979969718, −7.421201352292703252303805809983, −6.03779913047236765553795641399, −5.13053102890463408887667209519, −3.9680983777510616538695651785, −2.70418847945839487994115409771, −1.26471582840746602121440661680, 1.49905410106539184670477868869, 2.67737895649607083280227612435, 4.29420119251723896708360328797, 4.94527201719432427912956238526, 6.32709602680925752100354254548, 7.38793163811422583963802185430, 8.5214224383076908086469438603, 10.22362373547882841686853741730, 11.130339778626997298292680942826, 12.181221887061126086049300614306, 13.01798189476237298021328167981, 14.26977848491071948051110700304, 14.89598224626864559646170746341, 15.84319143533794741710923797847, 17.1298043461428817687187435971, 17.96960371681773000255637393418, 19.499363102803319496590684781556, 20.32473510127734976196499581321, 21.23717951707375884872727478566, 21.99232598329154129447632868125, 23.12799363256090236474693510741, 23.84374509630856685687239673780, 24.71896451064232958542575874801, 25.61570227991060272504110256821, 26.67367024298937840810585712454

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