Properties

Label 2-64400-1.1-c1-0-53
Degree $2$
Conductor $64400$
Sign $-1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 2·11-s + 13-s + 6·19-s − 21-s + 23-s − 5·27-s + 29-s − 31-s + 2·33-s + 6·37-s + 39-s + 3·41-s − 3·47-s + 49-s − 6·53-s + 6·57-s − 8·59-s − 10·61-s + 2·63-s − 2·67-s + 69-s − 5·71-s + 7·73-s − 2·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 1.37·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.185·29-s − 0.179·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s + 0.468·41-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.794·57-s − 1.04·59-s − 1.28·61-s + 0.251·63-s − 0.244·67-s + 0.120·69-s − 0.593·71-s + 0.819·73-s − 0.227·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.28927058214746, −14.13794232011178, −13.52719869286368, −13.11166969018195, −12.45874733050422, −11.98195016603039, −11.37998417771370, −11.11964367823845, −10.38427004723713, −9.622957581229150, −9.467768839088795, −8.861328161546193, −8.405914240792761, −7.695070697301613, −7.405882320614445, −6.633614041751502, −6.024405451704865, −5.692684091191265, −4.870397116429006, −4.292164889518952, −3.516892780479387, −3.107705501348945, −2.611242707494526, −1.679186209265867, −1.021681413555125, 0, 1.021681413555125, 1.679186209265867, 2.611242707494526, 3.107705501348945, 3.516892780479387, 4.292164889518952, 4.870397116429006, 5.692684091191265, 6.024405451704865, 6.633614041751502, 7.405882320614445, 7.695070697301613, 8.405914240792761, 8.861328161546193, 9.467768839088795, 9.622957581229150, 10.38427004723713, 11.11964367823845, 11.37998417771370, 11.98195016603039, 12.45874733050422, 13.11166969018195, 13.52719869286368, 14.13794232011178, 14.28927058214746

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