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Properties

Label 2-4600-5.4-c1-0-91
Degree $2$
Conductor $4600$
Sign $-0.894 + 0.447i$
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  + 1.33i·3-s − 3.16i·7-s + 1.21·9-s − 0.0955·11-s − 1.44i·13-s − 2.29i·17-s − 7.00·19-s + 4.23·21-s + i·23-s + 5.63i·27-s − 5.39·29-s − 0.584·31-s − 0.127i·33-s − 9.29i·37-s + 1.92·39-s + ⋯
L(s)  = 1  + 0.771i·3-s − 1.19i·7-s + 0.404·9-s − 0.0288·11-s − 0.400i·13-s − 0.557i·17-s − 1.60·19-s + 0.924·21-s + 0.208i·23-s + 1.08i·27-s − 1.00·29-s − 0.104·31-s − 0.0222i·33-s − 1.52i·37-s + 0.308·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 4600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3206186076\)
\(L(\frac12)\) \(\approx\) \(0.3206186076\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.33iT - 3T^{2} \)
7 \( 1 + 3.16iT - 7T^{2} \)
11 \( 1 + 0.0955T + 11T^{2} \)
13 \( 1 + 1.44iT - 13T^{2} \)
17 \( 1 + 2.29iT - 17T^{2} \)
19 \( 1 + 7.00T + 19T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
31 \( 1 + 0.584T + 31T^{2} \)
37 \( 1 + 9.29iT - 37T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
43 \( 1 + 9.50iT - 43T^{2} \)
47 \( 1 - 7.09iT - 47T^{2} \)
53 \( 1 - 7.73iT - 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 - 0.234T + 61T^{2} \)
67 \( 1 - 7.49iT - 67T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 - 1.52iT - 73T^{2} \)
79 \( 1 - 3.04T + 79T^{2} \)
83 \( 1 - 15.9iT - 83T^{2} \)
89 \( 1 + 5.53T + 89T^{2} \)
97 \( 1 - 2.58iT - 97T^{2} \)
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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

−7.87163363079916387928342007268, −7.33593371174570319737038441344, −6.67958676111943059188432425925, −5.70423872006258126089521521105, −4.90439370913317504484423439775, −4.02738829008157937484829692450, −3.83950131811420346671807450958, −2.58142245131451629952350080611, −1.39603915132822248243569456485, −0.083464861566401638230865893010, 1.60945118630869967200253306140, 2.09130403481138195476734195028, 3.11194983143618234860186896493, 4.21769881335244576035805386102, 4.92864370576408617886700730529, 6.00752583826197310171506586347, 6.37177983213710612225821831165, 7.09945136957869213035455556882, 8.035395336222544331150515818818, 8.469891212321689138089744325374

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