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Properties

Label 2-4600-5.4-c1-0-86
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  + 1.93i·3-s − 2.38i·7-s − 0.735·9-s + 5.33·11-s − 4.53i·13-s − 1.81i·17-s − 7.00·19-s + 4.60·21-s − i·23-s + 4.37i·27-s + 0.118·29-s − 0.884·31-s + 10.3i·33-s − 7.51i·37-s + 8.77·39-s + ⋯
L(s)  = 1  + 1.11i·3-s − 0.900i·7-s − 0.245·9-s + 1.60·11-s − 1.25i·13-s − 0.440i·17-s − 1.60·19-s + 1.00·21-s − 0.208i·23-s + 0.842i·27-s + 0.0219·29-s − 0.158·31-s + 1.79i·33-s − 1.23i·37-s + 1.40·39-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

\(L(1)\) \(\approx\) \(1.535654814\)
\(L(\frac12)\) \(\approx\) \(1.535654814\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93iT - 3T^{2} \)
7 \( 1 + 2.38iT - 7T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
13 \( 1 + 4.53iT - 13T^{2} \)
17 \( 1 + 1.81iT - 17T^{2} \)
19 \( 1 + 7.00T + 19T^{2} \)
29 \( 1 - 0.118T + 29T^{2} \)
31 \( 1 + 0.884T + 31T^{2} \)
37 \( 1 + 7.51iT - 37T^{2} \)
41 \( 1 + 1.45T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + 9.42iT - 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 + 2.80T + 61T^{2} \)
67 \( 1 - 3.11iT - 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 + 6.80T + 79T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 - 1.97iT - 97T^{2} \)
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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

−8.422678933663210852877007478803, −7.35217410902369788058936566199, −6.81150394179118088175065746996, −5.94135781883809971265270477857, −5.09398753846061180612977924342, −4.11353793816060064337813787024, −4.00127541698693301746154356237, −3.02305964704558438032843424720, −1.66935116697542982888828186182, −0.41896685777278383005656231709, 1.39094778762647276571077869526, 1.81678688265769728635427076265, 2.81260301332115250461559464842, 4.08333565496664150797820993493, 4.54346114852462180385377700555, 5.95714752263815074944169646016, 6.36079497887292761600896484336, 6.81088904856589700188092800629, 7.66851439934929594756191561269, 8.502179047248961302032576801037

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