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Properties

Label 2-33800-1.1-c1-0-9
Degree $2$
Conductor $33800$
Sign $1$
Analytic cond. $269.894$
Root an. cond. $16.4284$
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  + 3·3-s − 3·7-s + 6·9-s + 5·11-s − 3·17-s − 5·19-s − 9·21-s + 3·23-s + 9·27-s − 5·29-s + 8·31-s + 15·33-s + 9·37-s + 3·41-s + 43-s + 12·47-s + 2·49-s − 9·51-s − 2·53-s − 15·57-s − 5·59-s − 61-s − 18·63-s + 15·67-s + 9·69-s − 71-s + 10·73-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.13·7-s + 2·9-s + 1.50·11-s − 0.727·17-s − 1.14·19-s − 1.96·21-s + 0.625·23-s + 1.73·27-s − 0.928·29-s + 1.43·31-s + 2.61·33-s + 1.47·37-s + 0.468·41-s + 0.152·43-s + 1.75·47-s + 2/7·49-s − 1.26·51-s − 0.274·53-s − 1.98·57-s − 0.650·59-s − 0.128·61-s − 2.26·63-s + 1.83·67-s + 1.08·69-s − 0.118·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33800\)    =    \(2^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(269.894\)
Root analytic conductor: \(16.4284\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.650233944\)
\(L(\frac12)\) \(\approx\) \(4.650233944\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−15.00759699100888, −14.44822817034682, −13.94368148188025, −13.53423948195527, −12.90255106642094, −12.67349105722151, −11.99935316471654, −11.16937655344623, −10.70769873417364, −9.792210665346780, −9.478695825333847, −9.200431088676091, −8.530715967048950, −8.176046718530198, −7.341979619403438, −6.787586752257292, −6.447611868831511, −5.744342554538112, −4.450538157491739, −4.161266696405979, −3.637867365757016, −2.852465315027648, −2.460626585550672, −1.643707220191594, −0.7347785993194999, 0.7347785993194999, 1.643707220191594, 2.460626585550672, 2.852465315027648, 3.637867365757016, 4.161266696405979, 4.450538157491739, 5.744342554538112, 6.447611868831511, 6.787586752257292, 7.341979619403438, 8.176046718530198, 8.530715967048950, 9.200431088676091, 9.478695825333847, 9.792210665346780, 10.70769873417364, 11.16937655344623, 11.99935316471654, 12.67349105722151, 12.90255106642094, 13.53423948195527, 13.94368148188025, 14.44822817034682, 15.00759699100888

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