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Properties

Label 1-2652-2652.1283-r0-0-0
Degree $1$
Conductor $2652$
Sign $0.997 + 0.0694i$
Analytic cond. $12.3158$
Root an. cond. $12.3158$
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  + (−0.707 − 0.707i)5-s + (−0.258 − 0.965i)7-s + (0.965 + 0.258i)11-s + (−0.866 − 0.5i)19-s + (0.965 + 0.258i)23-s + i·25-s + (−0.258 + 0.965i)29-s + (0.707 + 0.707i)31-s + (−0.5 + 0.866i)35-s + (−0.258 + 0.965i)37-s + (0.965 + 0.258i)41-s + (0.866 + 0.5i)43-s − 47-s + (−0.866 + 0.5i)49-s + i·53-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (−0.258 − 0.965i)7-s + (0.965 + 0.258i)11-s + (−0.866 − 0.5i)19-s + (0.965 + 0.258i)23-s + i·25-s + (−0.258 + 0.965i)29-s + (0.707 + 0.707i)31-s + (−0.5 + 0.866i)35-s + (−0.258 + 0.965i)37-s + (0.965 + 0.258i)41-s + (0.866 + 0.5i)43-s − 47-s + (−0.866 + 0.5i)49-s + i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $0.997 + 0.0694i$
Analytic conductor: \(12.3158\)
Root analytic conductor: \(12.3158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (0:\ ),\ 0.997 + 0.0694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.315396612 + 0.04575952418i\)
\(L(\frac12)\) \(\approx\) \(1.315396612 + 0.04575952418i\)
\(L(1)\) \(\approx\) \(0.9558347027 - 0.1156983614i\)
\(L(1)\) \(\approx\) \(0.9558347027 - 0.1156983614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (-0.258 + 0.965i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (0.965 + 0.258i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.965 - 0.258i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
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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

−19.28405082122322888256161268876, −18.805476674397917242036807559808, −18.03472079995428631872976871781, −17.16311812510449415613579181732, −16.48333409482463289173370173397, −15.602159744146762108935719833424, −15.107099420531893682429980921031, −14.50796023349895298169456105499, −13.766268079635933291197517338567, −12.60694001006962453110874713068, −12.25822702223851975469129538759, −11.28556630364766502077463470264, −10.981917946747922816681294485499, −9.837430286079864098274480251242, −9.17969975428519147874415342327, −8.374768300202879105969911913477, −7.72576367524411293701286255020, −6.626708424606344213297896873562, −6.30394475259751396656379492592, −5.334914882527979725426656722125, −4.2128054108020628280533551722, −3.64357132195623894439107129154, −2.69627277859474831559598723162, −1.99730726756746556676682795798, −0.55424525342520255782143276644, 0.8773258899307460895403435826, 1.47713779749189421564277725438, 2.92561898874834485880695710989, 3.73037051985373282359340945421, 4.4586759812691382553824579296, 4.97387247093809374673495403330, 6.2622265319921771456968208726, 6.94331473794688928751033981559, 7.57687302083744657035819862375, 8.51302157009476687164081656663, 9.114459780857494940378102598775, 9.876164743384955783729945402470, 10.88309215842052815697658481840, 11.339282115245375155564866089525, 12.3273247487791538604939109601, 12.817404700493938623018263354834, 13.56945826049315493123924219300, 14.40808404422697089264566524193, 15.09522782443598058663725363716, 15.89213960403118195374258148335, 16.589754075240891264929459785736, 17.14265606983630494537151559141, 17.65973625225320776919206723078, 18.88183779991792586553294648761, 19.538811307200653966610070090670

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