↑

Properties

Label 1-312-312.77-r1-0-0
Degree $1$
Conductor $312$
Sign $1$
Analytic cond. $33.5290$
Root an. cond. $33.5290$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 7-s − 11-s − 17-s + 19-s − 23-s + 25-s + 29-s − 31-s + 35-s + 37-s + 41-s − 43-s + 47-s + 49-s + 53-s + 55-s − 59-s − 61-s + 67-s + 71-s − 73-s + 77-s + 79-s − 83-s + 85-s + 89-s + ⋯
L(s)  = 1  − 5-s − 7-s − 11-s − 17-s + 19-s − 23-s + 25-s + 29-s − 31-s + 35-s + 37-s + 41-s − 43-s + 47-s + 49-s + 53-s + 55-s − 59-s − 61-s + 67-s + 71-s − 73-s + 77-s + 79-s − 83-s + 85-s + 89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(33.5290\)
Root analytic conductor: \(33.5290\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{312} (77, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 312,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8567756566\)
\(L(\frac12)\) \(\approx\) \(0.8567756566\)
\(L(1)\) \(\approx\) \(0.7114306424\)
\(L(1)\) \(\approx\) \(0.7114306424\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.96827020055537323318557457607, −23.965749095167798692105806614062, −23.253822539072021816297900565949, −22.43790158013381623868498301154, −21.57546785859863936777791645505, −20.08211155949000679242230339609, −19.91275962128752176405038067648, −18.66575367284540415350126910230, −18.02837084051318966759288178710, −16.54064349784720860350836847410, −15.87489918715940892590013707210, −15.29054446297885888925453805326, −13.915909147539468823612799418443, −12.95727321967270526047581568702, −12.14202132042952269326732075440, −11.0996126649592096530508104667, −10.13160634141101693855109768314, −9.03386023828865552447535068227, −7.92498113496387382092840922285, −7.08548337500008305305870067246, −5.92217525167422673657713192972, −4.62533886719760649489880360996, −3.53936340174753457794290489655, −2.52206530120099865429565649861, −0.52715380629490367684074550437, 0.52715380629490367684074550437, 2.52206530120099865429565649861, 3.53936340174753457794290489655, 4.62533886719760649489880360996, 5.92217525167422673657713192972, 7.08548337500008305305870067246, 7.92498113496387382092840922285, 9.03386023828865552447535068227, 10.13160634141101693855109768314, 11.0996126649592096530508104667, 12.14202132042952269326732075440, 12.95727321967270526047581568702, 13.915909147539468823612799418443, 15.29054446297885888925453805326, 15.87489918715940892590013707210, 16.54064349784720860350836847410, 18.02837084051318966759288178710, 18.66575367284540415350126910230, 19.91275962128752176405038067648, 20.08211155949000679242230339609, 21.57546785859863936777791645505, 22.43790158013381623868498301154, 23.253822539072021816297900565949, 23.965749095167798692105806614062, 24.96827020055537323318557457607

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