Properties

Label 2-4600-5.4-c1-0-8
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  − 2.66i·3-s + 3.66i·7-s − 4.12·9-s − 1.21·11-s − 2.21i·13-s + 1.21i·17-s + 2.57·19-s + 9.79·21-s i·23-s + 3.00i·27-s − 1.45·29-s − 6.46·31-s + 3.24i·33-s + 4i·37-s − 5.90·39-s + ⋯
L(s)  = 1  − 1.54i·3-s + 1.38i·7-s − 1.37·9-s − 0.366·11-s − 0.614i·13-s + 0.294i·17-s + 0.591·19-s + 2.13·21-s − 0.208i·23-s + 0.577i·27-s − 0.270·29-s − 1.16·31-s + 0.564i·33-s + 0.657i·37-s − 0.946·39-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

\(L(1)\) \(\approx\) \(0.8053561500\)
\(L(\frac12)\) \(\approx\) \(0.8053561500\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.66iT - 3T^{2} \)
7 \( 1 - 3.66iT - 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + 2.21iT - 13T^{2} \)
17 \( 1 - 1.21iT - 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 6.90iT - 43T^{2} \)
47 \( 1 - 5.45iT - 47T^{2} \)
53 \( 1 + 3.81iT - 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 - 12.8iT - 67T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 - 15.2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 1.69iT - 97T^{2} \)
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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

−8.401774413761988900295450274876, −7.71427249953301410615493773767, −7.01369410062183134363355799737, −6.35437238795840323404033792350, −5.50475642251607866870889554286, −5.24373888982534680842423255941, −3.67604046992134959270165361614, −2.69490317863187618736973825999, −2.13165625011420167762150851159, −1.15457487375190425493734328003, 0.22755130811372979630780094344, 1.73696861946420270498146359549, 3.19504905852328716046422497621, 3.68186825769584694207014936663, 4.44514158840672353155207747066, 4.99330874192870947658213427225, 5.77602099185594838681160732704, 6.87568316066348355385048410985, 7.43176737154185137243283328064, 8.263698512240336105506373485944

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