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Properties

Label 2-34650-1.1-c1-0-40
Degree $2$
Conductor $34650$
Sign $1$
Analytic cond. $276.681$
Root an. cond. $16.6337$
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 + 4-s − 7-s + 8-s + 11-s + 4·13-s − 14-s + 16-s − 4·19-s + 22-s + 4·26-s − 28-s + 6·29-s − 10·31-s + 32-s − 2·37-s − 4·38-s + 12·41-s + 4·43-s + 44-s + 6·47-s + 49-s + 4·52-s − 6·53-s − 56-s + 6·58-s + 6·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 0.301·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.213·22-s + 0.784·26-s − 0.188·28-s + 1.11·29-s − 1.79·31-s + 0.176·32-s − 0.328·37-s − 0.648·38-s + 1.87·41-s + 0.609·43-s + 0.150·44-s + 0.875·47-s + 1/7·49-s + 0.554·52-s − 0.824·53-s − 0.133·56-s + 0.787·58-s + 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(34650\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(276.681\)
Root analytic conductor: \(16.6337\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 34650,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.828103049\)
\(L(\frac12)\) \(\approx\) \(3.828103049\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.91555443007619, −14.23274439277464, −14.09275999660240, −13.28744567104585, −12.89181150214366, −12.47676092733315, −11.91608588515713, −11.22016400250661, −10.74267107598515, −10.48964553836762, −9.508014642068758, −9.102028840465362, −8.499411191731756, −7.866250736101080, −7.186670248727971, −6.642446924242284, −6.029971041189450, −5.716306537379260, −4.869033378646049, −4.128217712839053, −3.805541923145240, −3.039944981409309, −2.335160706137482, −1.555076575074961, −0.6585681453153803, 0.6585681453153803, 1.555076575074961, 2.335160706137482, 3.039944981409309, 3.805541923145240, 4.128217712839053, 4.869033378646049, 5.716306537379260, 6.029971041189450, 6.642446924242284, 7.186670248727971, 7.866250736101080, 8.499411191731756, 9.102028840465362, 9.508014642068758, 10.48964553836762, 10.74267107598515, 11.22016400250661, 11.91608588515713, 12.47676092733315, 12.89181150214366, 13.28744567104585, 14.09275999660240, 14.23274439277464, 14.91555443007619

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