Properties

Label 2-4600-5.4-c1-0-41
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  + 3i·3-s − 2i·7-s − 6·9-s i·13-s + 6·21-s i·23-s − 9i·27-s + 3·29-s + 3·31-s − 8i·37-s + 3·39-s + 3·41-s + 2i·43-s − 11i·47-s + 3·49-s + ⋯
L(s)  = 1  + 1.73i·3-s − 0.755i·7-s − 2·9-s − 0.277i·13-s + 1.30·21-s − 0.208i·23-s − 1.73i·27-s + 0.557·29-s + 0.538·31-s − 1.31i·37-s + 0.480·39-s + 0.468·41-s + 0.304i·43-s − 1.60i·47-s + 0.428·49-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

\(L(1)\) \(\approx\) \(1.731287754\)
\(L(\frac12)\) \(\approx\) \(1.731287754\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 11iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−8.662486920376952591533446037602, −7.85205156031396279168738253681, −7.02404820238043737778499722383, −6.04178741119857415136783117697, −5.34321068353149304637587292023, −4.57194838248218024187446697513, −4.01931324585700801304247440971, −3.34227735495195328919822928698, −2.40255080554156027422688666389, −0.69843220659036803517831161922, 0.76528491901308749605670720691, 1.76183632493648659148647100219, 2.49210468537196315234766297334, 3.27879554403493564280515281657, 4.61060177125585013665299574561, 5.51064474057715498178683223258, 6.22224997531827549575503713729, 6.70273101460077679967451138801, 7.45314201601922274236887331474, 8.149097666441831610179014039050

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