Properties

Label 2-64400-1.1-c1-0-34
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  − 7-s − 3·9-s − 2·11-s + 4·13-s − 6·17-s − 23-s + 2·29-s + 2·31-s + 4·37-s − 2·41-s − 4·43-s + 49-s + 2·59-s + 2·61-s + 3·63-s + 8·67-s + 16·71-s − 4·73-s + 2·77-s − 10·79-s + 9·81-s + 4·83-s − 10·89-s − 4·91-s − 2·97-s + 6·99-s + 101-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 0.603·11-s + 1.10·13-s − 1.45·17-s − 0.208·23-s + 0.371·29-s + 0.359·31-s + 0.657·37-s − 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.260·59-s + 0.256·61-s + 0.377·63-s + 0.977·67-s + 1.89·71-s − 0.468·73-s + 0.227·77-s − 1.12·79-s + 81-s + 0.439·83-s − 1.05·89-s − 0.419·91-s − 0.203·97-s + 0.603·99-s + 0.0995·101-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.35326442122323, −13.90566173804388, −13.50828644686073, −13.03331620045806, −12.59523222595079, −11.87138519237100, −11.33918099345219, −11.04488457018711, −10.53871347975726, −9.872447048841189, −9.376543779225234, −8.681627799304708, −8.407664929845611, −7.971623465191425, −7.103454592091118, −6.540972897611105, −6.196415148109781, −5.529158009417965, −5.032244346101494, −4.255187317470019, −3.741847157579630, −2.975943508395049, −2.530139669857436, −1.796523546637693, −0.7848975743528048, 0, 0.7848975743528048, 1.796523546637693, 2.530139669857436, 2.975943508395049, 3.741847157579630, 4.255187317470019, 5.032244346101494, 5.529158009417965, 6.196415148109781, 6.540972897611105, 7.103454592091118, 7.971623465191425, 8.407664929845611, 8.681627799304708, 9.376543779225234, 9.872447048841189, 10.53871347975726, 11.04488457018711, 11.33918099345219, 11.87138519237100, 12.59523222595079, 13.03331620045806, 13.50828644686073, 13.90566173804388, 14.35326442122323

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