Properties

Label 2-338100-1.1-c1-0-8
Degree $2$
Conductor $338100$
Sign $1$
Analytic cond. $2699.74$
Root an. cond. $51.9590$
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-s + 9-s − 11-s + 4·13-s + 6·17-s − 7·19-s + 23-s − 27-s − 6·29-s − 4·31-s + 33-s + 2·37-s − 4·39-s − 9·41-s − 2·43-s + 7·47-s − 6·51-s − 5·53-s + 7·57-s − 7·59-s + 7·61-s + 2·67-s − 69-s + 4·71-s − 10·73-s − 6·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.45·17-s − 1.60·19-s + 0.208·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 1.40·41-s − 0.304·43-s + 1.02·47-s − 0.840·51-s − 0.686·53-s + 0.927·57-s − 0.911·59-s + 0.896·61-s + 0.244·67-s − 0.120·69-s + 0.474·71-s − 1.17·73-s − 0.675·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2699.74\)
Root analytic conductor: \(51.9590\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338100} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338100,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.184371218\)
\(L(\frac12)\) \(\approx\) \(1.184371218\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−12.66607600158088, −12.04098921304145, −11.73695227901209, −11.10172423744492, −10.79723920328481, −10.44107981520189, −9.957697216981817, −9.399715362674537, −8.929858763708406, −8.409051639173460, −8.004712676017556, −7.496551622492678, −6.983683105198493, −6.423652811111166, −6.047511921865927, −5.526130735740888, −5.206259453585692, −4.517664591885914, −3.951856884641765, −3.558743907023993, −3.018949442200952, −2.199021625728533, −1.657506566569528, −1.123567430898201, −0.3130307407169617, 0.3130307407169617, 1.123567430898201, 1.657506566569528, 2.199021625728533, 3.018949442200952, 3.558743907023993, 3.951856884641765, 4.517664591885914, 5.206259453585692, 5.526130735740888, 6.047511921865927, 6.423652811111166, 6.983683105198493, 7.496551622492678, 8.004712676017556, 8.409051639173460, 8.929858763708406, 9.399715362674537, 9.957697216981817, 10.44107981520189, 10.79723920328481, 11.10172423744492, 11.73695227901209, 12.04098921304145, 12.66607600158088

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