Properties

Label 2-33930-1.1-c1-0-15
Degree $2$
Conductor $33930$
Sign $1$
Analytic cond. $270.932$
Root an. cond. $16.4600$
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 − 5-s + 8-s − 10-s + 4·11-s − 13-s + 16-s + 4·17-s − 20-s + 4·22-s − 2·23-s + 25-s − 26-s + 29-s + 4·31-s + 32-s + 4·34-s + 2·37-s − 40-s + 4·43-s + 4·44-s − 2·46-s − 7·49-s + 50-s − 52-s + 2·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.852·22-s − 0.417·23-s + 1/5·25-s − 0.196·26-s + 0.185·29-s + 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.328·37-s − 0.158·40-s + 0.609·43-s + 0.603·44-s − 0.294·46-s − 49-s + 0.141·50-s − 0.138·52-s + 0.274·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 29\)
Sign: $1$
Analytic conductor: \(270.932\)
Root analytic conductor: \(16.4600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.149503918\)
\(L(\frac12)\) \(\approx\) \(4.149503918\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
13 \( 1 + T \)
29 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.89728329276228, −14.44093148687371, −14.01293190090984, −13.54071993306900, −12.72522343864599, −12.35064259113274, −11.93847078185560, −11.37520632552948, −10.99728973791424, −10.10977617374216, −9.767732931034814, −9.120617613822378, −8.331783600626399, −7.939259605938023, −7.243473615177966, −6.664852814939581, −6.219505767238465, −5.474845153354829, −4.930570683901356, −4.137329687850969, −3.811253930414095, −3.090069451997605, −2.369618597883099, −1.460665647656543, −0.7191884788527298, 0.7191884788527298, 1.460665647656543, 2.369618597883099, 3.090069451997605, 3.811253930414095, 4.137329687850969, 4.930570683901356, 5.474845153354829, 6.219505767238465, 6.664852814939581, 7.243473615177966, 7.939259605938023, 8.331783600626399, 9.120617613822378, 9.767732931034814, 10.10977617374216, 10.99728973791424, 11.37520632552948, 11.93847078185560, 12.35064259113274, 12.72522343864599, 13.54071993306900, 14.01293190090984, 14.44093148687371, 14.89728329276228

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