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Properties

Label 1-300-300.203-r1-0-0
Degree $1$
Conductor $300$
Sign $0.929 - 0.368i$
Analytic cond. $32.2394$
Root an. cond. $32.2394$
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·7-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.587 − 0.809i)23-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.587 + 0.809i)37-s + (0.809 − 0.587i)41-s − i·43-s + (0.951 − 0.309i)47-s − 49-s + (0.951 − 0.309i)53-s + (0.809 − 0.587i)59-s + ⋯
L(s)  = 1  − i·7-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.587 − 0.809i)23-s + (0.309 + 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.587 + 0.809i)37-s + (0.809 − 0.587i)41-s − i·43-s + (0.951 − 0.309i)47-s − 49-s + (0.951 − 0.309i)53-s + (0.809 − 0.587i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(32.2394\)
Root analytic conductor: \(32.2394\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (1:\ ),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.708712869 - 0.3259544125i\)
\(L(\frac12)\) \(\approx\) \(1.708712869 - 0.3259544125i\)
\(L(1)\) \(\approx\) \(1.082052389 + 0.02517431269i\)
\(L(1)\) \(\approx\) \(1.082052389 + 0.02517431269i\)

Euler product

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

−25.25187962594722945478601250965, −24.22375404434321154334940497050, −23.25606146183929363229944298331, −22.840443739615053386084151948609, −21.37155239209129828150901441416, −20.80712843927554294975779384540, −19.80232566290063980116615532157, −18.91278792392076753506891485722, −17.91519865989406225975557675747, −16.90443543458408325129480398506, −16.26272720400417940181387173627, −15.01570247354021562752067028017, −14.14289464993808416230093799890, −13.25028909710313175938960163653, −12.22413389061134091597949287469, −11.19363829817215277332256951803, −10.07945177422234661767436020391, −9.46165109947423417921398293251, −7.74572441232948434013044438449, −7.3935598192719387135183241084, −5.95453979118916411966877832641, −4.76137142820051703643467249044, −3.76773484455578889752555444698, −2.37060858985301181848741860438, −0.93622791569917980011859760954, 0.68248026847244654193117365062, 2.44685350317132873427585060876, 3.24876673839651071353201823635, 5.0354217308601448831416785172, 5.60428606056776430126791794113, 6.96860797080595339103451504598, 8.13735005904155092954517658755, 8.93857992527078132480182478420, 10.148590156048419548588304462713, 11.03908351111907036891758448944, 12.26779239583939989412350355635, 12.8662809988502255613751455778, 14.11806592941448152169389045688, 15.10698966154741227506972935391, 15.83411038487972464217326532634, 16.87067515320972233695746080913, 17.998754458471821883867104604358, 18.67555596484302950844199628153, 19.60678287411333617118549649879, 20.67413022097257067397156609544, 21.61505680197442800056152534873, 22.22611407094206241242212041843, 23.37798889172789739129326928492, 24.23606310448495075718504581496, 25.1102504711718286800804929166

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