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Properties

Label 1-300-300.239-r0-0-0
Degree $1$
Conductor $300$
Sign $0.876 - 0.481i$
Analytic cond. $1.39319$
Root an. cond. $1.39319$
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  + 7-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.309 + 0.951i)23-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.309 − 0.951i)37-s + (−0.309 − 0.951i)41-s + 43-s + (0.809 + 0.587i)47-s + 49-s + (−0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s + ⋯
L(s)  = 1  + 7-s + (0.309 − 0.951i)11-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.309 + 0.951i)23-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.309 − 0.951i)37-s + (−0.309 − 0.951i)41-s + 43-s + (0.809 + 0.587i)47-s + 49-s + (−0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(1.39319\)
Root analytic conductor: \(1.39319\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (0:\ ),\ 0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.334073470 - 0.3425318488i\)
\(L(\frac12)\) \(\approx\) \(1.334073470 - 0.3425318488i\)
\(L(1)\) \(\approx\) \(1.166433769 - 0.1304756859i\)
\(L(1)\) \(\approx\) \(1.166433769 - 0.1304756859i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−25.23842068897265483803529031469, −24.59187666916484048765203363924, −23.73372504154675362958528045779, −22.72580284878510354141183121114, −21.88880987197735897411023595932, −20.799176115936774161868326369975, −20.24618419798273313003202660201, −19.07574101559817175913851628862, −18.05937893839914809062419383727, −17.41185262475724827152788888063, −16.346494191006991002710151664398, −15.29333981778196014747019651620, −14.35750535557227472703129890591, −13.68712459268936569650898800724, −12.16537312218122341020772676077, −11.70373833535843750257657119827, −10.4633722458407544555774692749, −9.458933691169520995990547561911, −8.41050100771116880117609758134, −7.35410659745524512276991861921, −6.401666730823058345532717272, −4.86869674353155220284837165557, −4.31198378244156813036223647123, −2.56304034518901371038281317060, −1.46470704591666639129183716630, 1.065897994396869153328904278795, 2.52670795838378273753357247267, 3.81677782723715960415287067891, 5.05101908521031504582386046837, 5.95864723081911255155971988693, 7.3376885614602324026173789907, 8.2500567654696748565650778639, 9.15525968919574760061581322021, 10.51858050321023181718320664747, 11.25886538177266839510850284856, 12.212666624571937107085821627090, 13.45516052531987715568272914076, 14.191912811728625017104532786383, 15.26413308685689261220527006414, 16.04249276500130988091712925344, 17.52320505284154110489301559331, 17.667780984148105818736406063650, 19.062103220012609491557771563629, 19.88839958462440285486884558516, 20.81787351146970264088956370595, 21.77910550862278305924802142898, 22.42703868060732339970513942300, 23.77833703442257472913521953195, 24.31751605445984187185709927958, 25.136016226715795442830153018541

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