Properties

Label 2-4600-5.4-c1-0-55
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  − 2.14i·3-s + 1.14i·7-s − 1.60·9-s + 5.89·11-s − 4.89i·13-s + 5.89i·17-s + 2.34·19-s + 2.45·21-s + i·23-s − 3.00i·27-s − 3.74·29-s + 5.68·31-s − 12.6i·33-s − 4i·37-s − 10.4·39-s + ⋯
L(s)  = 1  − 1.23i·3-s + 0.432i·7-s − 0.533·9-s + 1.77·11-s − 1.35i·13-s + 1.42i·17-s + 0.538·19-s + 0.536·21-s + 0.208i·23-s − 0.577i·27-s − 0.695·29-s + 1.02·31-s − 2.20i·33-s − 0.657i·37-s − 1.68·39-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

\(L(1)\) \(\approx\) \(2.353869447\)
\(L(\frac12)\) \(\approx\) \(2.353869447\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.14iT - 3T^{2} \)
7 \( 1 - 1.14iT - 7T^{2} \)
11 \( 1 - 5.89T + 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 - 5.89iT - 17T^{2} \)
19 \( 1 - 2.34T + 19T^{2} \)
29 \( 1 + 3.74T + 29T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 1.05T + 41T^{2} \)
43 \( 1 - 11.4iT - 43T^{2} \)
47 \( 1 + 7.74iT - 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + 0.797T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 15.5iT - 67T^{2} \)
71 \( 1 - 2.94T + 71T^{2} \)
73 \( 1 - 6.32iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 0.912iT - 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
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

−8.214695765612354827721051065873, −7.39041522636624481299074977858, −6.80555748925419487343258283847, −5.98593536459781710849982527412, −5.68450864140631065343569187143, −4.33333153287288943236617017286, −3.56994557156115374059874746751, −2.56279500520851584166569270445, −1.55019230782408472121306907292, −0.907261291625832956941011649745, 0.937625008117575699146003111847, 2.13256248045723337378687682754, 3.49234594401900469184531078792, 3.88581515319703185783351045167, 4.66656246090835631506269127345, 5.19027630655108844096440801191, 6.47229399241828668527458561107, 6.81778489746481608507098660483, 7.67001691405298536189260137580, 8.876991173894774500159533445813

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