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Properties

Label 1-312-312.203-r1-0-0
Degree $1$
Conductor $312$
Sign $-0.957 + 0.289i$
Analytic cond. $33.5290$
Root an. cond. $33.5290$
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  + i·5-s + i·7-s + i·11-s + 17-s + i·19-s − 23-s − 25-s + 29-s − i·31-s − 35-s + i·37-s − i·41-s − 43-s − i·47-s − 49-s + ⋯
L(s)  = 1  + i·5-s + i·7-s + i·11-s + 17-s + i·19-s − 23-s − 25-s + 29-s − i·31-s − 35-s + i·37-s − i·41-s − 43-s − i·47-s − 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(33.5290\)
Root analytic conductor: \(33.5290\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 312,\ (1:\ ),\ -0.957 + 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1970239557 + 1.330624930i\)
\(L(\frac12)\) \(\approx\) \(0.1970239557 + 1.330624930i\)
\(L(1)\) \(\approx\) \(0.8870659183 + 0.4747432722i\)
\(L(1)\) \(\approx\) \(0.8870659183 + 0.4747432722i\)

Euler product

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

−24.457624467484152730073182378812, −23.754103914457849184507064099883, −23.08851241756346649093757839857, −21.665049202166002585471881881177, −21.112763051876078841307618406574, −19.93365802920637572657940051751, −19.57410755676338856738290760837, −18.1809492982627772488089865725, −17.19947349810513842998427005030, −16.43777624947088423770600956550, −15.804711010215685767421742391659, −14.257273861425397046592482780470, −13.63242575919523319165970801018, −12.67328139394095819150256897917, −11.66869134561558153081956292097, −10.606224406141971955187950939810, −9.61231306376581690431313074134, −8.51357369405763771262933027030, −7.7170889321847483625874554921, −6.42866887967800756864583908545, −5.26836480549529054803367817097, −4.27294031951085854068091643498, −3.15615960319970123228280529631, −1.36594921484768240074600523667, −0.407506677957922296541724872783, 1.79795765234518023656965218358, 2.79430297154271108487029861069, 3.987028124104033747858826024588, 5.44143780397832806865501489686, 6.3235566965257416609442234742, 7.44260032875396166053216613448, 8.38822309631212583886974735976, 9.810210710558454488452455811662, 10.295108056983140133122163123173, 11.81888465524557072188035494429, 12.18346303470290243074633791836, 13.61289323618944285503300523565, 14.68326443512631601102642852597, 15.16378815723864409816404092901, 16.23336702239108137026599189594, 17.457581490289866245975991687026, 18.39782204243165710556403671762, 18.836529907855625253451806687373, 20.00236114653010043124592840293, 21.07390694465513588010877391248, 21.9226766252743457965126436093, 22.71550893678061215233623854633, 23.42170607918654194589882801289, 24.69848298168995172008893930769, 25.550003369213712263342812706940

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