Properties

Label 2-4600-5.4-c1-0-47
Degree $2$
Conductor $4600$
Sign $0.447 + 0.894i$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56i·3-s − 3.56·9-s + 5.12·11-s + 4.56i·13-s + 3.12i·17-s − 5.12·19-s i·23-s + 1.43i·27-s + 0.561·29-s − 6.56·31-s − 13.1i·33-s + 8.24i·37-s + 11.6·39-s + 10.8·41-s − 8i·43-s + ⋯
L(s)  = 1  − 1.47i·3-s − 1.18·9-s + 1.54·11-s + 1.26i·13-s + 0.757i·17-s − 1.17·19-s − 0.208i·23-s + 0.276i·27-s + 0.104·29-s − 1.17·31-s − 2.28i·33-s + 1.35i·37-s + 1.87·39-s + 1.68·41-s − 1.21i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.025746643\)
\(L(\frac12)\) \(\approx\) \(2.025746643\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + iT \)
good3 \( 1 + 2.56iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 - 4.56iT - 13T^{2} \)
17 \( 1 - 3.12iT - 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
29 \( 1 - 0.561T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 - 8.24iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 5.12iT - 67T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 + 2.31iT - 73T^{2} \)
79 \( 1 - 5.12T + 79T^{2} \)
83 \( 1 + 2.24iT - 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.223194188839842677286287051048, −7.23161536392081926461291779014, −6.70418368249559564078088879173, −6.39493552824126593723961247281, −5.51122781450979889972416021916, −4.21486758469910550504300723464, −3.77619244735410390382509217800, −2.24743638525632042165945964563, −1.81316568199707686160866374596, −0.814681270084052902003881604503, 0.800811225401481398194365439291, 2.31593614068389934968889262050, 3.35194577839899351848530003943, 3.98123345621833431160936344067, 4.55329600351763514626853545604, 5.46396278588778067804249992353, 6.03914232563981520678430201985, 6.98073231382581887409942477445, 7.82992079831242737983847822265, 8.696383306322974671358961021601

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