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Properties

Label 2-420e2-1.1-c1-0-19
Degree $2$
Conductor $176400$
Sign $1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
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  − 2·13-s − 8·19-s + 7·31-s − 11·37-s + 5·43-s − 61-s − 16·67-s − 17·73-s − 17·79-s + 19·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.554·13-s − 1.83·19-s + 1.25·31-s − 1.80·37-s + 0.762·43-s − 0.128·61-s − 1.95·67-s − 1.98·73-s − 1.91·79-s + 1.92·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6560786984\)
\(L(\frac12)\) \(\approx\) \(0.6560786984\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−13.04884910823048, −12.82373018546221, −12.13515742023423, −11.90379688623335, −11.34327460591208, −10.67578001215961, −10.25648546562117, −10.15948505528740, −9.216328783895179, −8.902326415620650, −8.446908745939724, −7.949900671437174, −7.293439381386460, −6.960077862045043, −6.293873136709034, −5.959570605051865, −5.312155227952428, −4.602771212499465, −4.386043638248872, −3.720384849299267, −2.971458224074018, −2.532497071863597, −1.851344500746912, −1.272506344221555, −0.2290410806961714, 0.2290410806961714, 1.272506344221555, 1.851344500746912, 2.532497071863597, 2.971458224074018, 3.720384849299267, 4.386043638248872, 4.602771212499465, 5.312155227952428, 5.959570605051865, 6.293873136709034, 6.960077862045043, 7.293439381386460, 7.949900671437174, 8.446908745939724, 8.902326415620650, 9.216328783895179, 10.15948505528740, 10.25648546562117, 10.67578001215961, 11.34327460591208, 11.90379688623335, 12.13515742023423, 12.82373018546221, 13.04884910823048

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