Properties

Degree $2$
Conductor $193600$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·7-s + 9-s + 5·13-s − 3·17-s − 2·19-s + 4·21-s − 6·23-s + 4·27-s + 3·29-s + 2·31-s − 7·37-s − 10·39-s + 3·41-s + 8·43-s − 6·47-s − 3·49-s + 6·51-s − 3·53-s + 4·57-s + 10·61-s − 2·63-s − 10·67-s + 12·69-s + 12·71-s − 14·73-s + 2·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.38·13-s − 0.727·17-s − 0.458·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s + 0.557·29-s + 0.359·31-s − 1.15·37-s − 1.60·39-s + 0.468·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.412·53-s + 0.529·57-s + 1.28·61-s − 0.251·63-s − 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.63·73-s + 0.225·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{193600} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 193600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5075359973\)
\(L(\frac12)\) \(\approx\) \(0.5075359973\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−12.97705937721296, −12.59110975886633, −12.06634837111968, −11.75724438474453, −11.11943811302052, −10.80440095651926, −10.46921666582825, −9.879059956380801, −9.368907857610621, −8.828058696677816, −8.280273164888436, −7.996458655376876, −7.032687516965832, −6.661278651471409, −6.282181113921611, −5.899899616434919, −5.441561635244378, −4.769689334881846, −4.194996014700448, −3.776965874827062, −3.095971983191217, −2.472715029638035, −1.697418954985647, −1.026359801206637, −0.2487693461631769, 0.2487693461631769, 1.026359801206637, 1.697418954985647, 2.472715029638035, 3.095971983191217, 3.776965874827062, 4.194996014700448, 4.769689334881846, 5.441561635244378, 5.899899616434919, 6.282181113921611, 6.661278651471409, 7.032687516965832, 7.996458655376876, 8.280273164888436, 8.828058696677816, 9.368907857610621, 9.879059956380801, 10.46921666582825, 10.80440095651926, 11.11943811302052, 11.75724438474453, 12.06634837111968, 12.59110975886633, 12.97705937721296

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