Properties

Label 2-61710-1.1-c1-0-40
Degree $2$
Conductor $61710$
Sign $1$
Analytic cond. $492.756$
Root an. cond. $22.1981$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s − 2·13-s + 4·14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 4·21-s + 24-s + 25-s − 2·26-s + 27-s + 4·28-s + 6·29-s + 30-s + 8·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61710\)    =    \(2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(492.756\)
Root analytic conductor: \(22.1981\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{61710} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 61710,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.711786560\)
\(L(\frac12)\) \(\approx\) \(8.711786560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.18874022007591, −13.78317464236013, −13.66559070087424, −12.77780311603154, −12.20927599969465, −11.84393182443587, −11.45371839839350, −10.68270434513970, −10.14142962228229, −9.961550898864896, −8.908339652569029, −8.628457109384491, −8.061488174828707, −7.394644190630757, −7.129797891472181, −6.363935763816548, −5.624845582752910, −5.115605491962071, −4.765034335333403, −4.113869428255549, −3.446715899367254, −2.634130500702864, −2.290675306273740, −1.458567391265456, −0.9224688737259843, 0.9224688737259843, 1.458567391265456, 2.290675306273740, 2.634130500702864, 3.446715899367254, 4.113869428255549, 4.765034335333403, 5.115605491962071, 5.624845582752910, 6.363935763816548, 7.129797891472181, 7.394644190630757, 8.061488174828707, 8.628457109384491, 8.908339652569029, 9.961550898864896, 10.14142962228229, 10.68270434513970, 11.45371839839350, 11.84393182443587, 12.20927599969465, 12.77780311603154, 13.66559070087424, 13.78317464236013, 14.18874022007591

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