Properties

Label 2-3200-1.1-c1-0-13
Degree $2$
Conductor $3200$
Sign $1$
Analytic cond. $25.5521$
Root an. cond. $5.05491$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·3-s − 1.23·7-s + 7.47·9-s + 2·11-s + 4.47·13-s + 4.47·17-s − 4.47·19-s + 4.00·21-s + 9.23·23-s − 14.4·27-s + 2·29-s − 2.47·31-s − 6.47·33-s − 10.9·37-s − 14.4·39-s + 3.52·41-s + 5.70·43-s + 2.76·47-s − 5.47·49-s − 14.4·51-s − 8.47·53-s + 14.4·57-s − 0.472·59-s + 6·61-s − 9.23·63-s − 5.70·67-s − 29.8·69-s + ⋯
L(s)  = 1  − 1.86·3-s − 0.467·7-s + 2.49·9-s + 0.603·11-s + 1.24·13-s + 1.08·17-s − 1.02·19-s + 0.872·21-s + 1.92·23-s − 2.78·27-s + 0.371·29-s − 0.444·31-s − 1.12·33-s − 1.79·37-s − 2.31·39-s + 0.550·41-s + 0.870·43-s + 0.403·47-s − 0.781·49-s − 2.02·51-s − 1.16·53-s + 1.91·57-s − 0.0614·59-s + 0.768·61-s − 1.16·63-s − 0.697·67-s − 3.59·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(25.5521\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.002857952\)
\(L(\frac12)\) \(\approx\) \(1.002857952\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + 3.23T + 3T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 + 4.47T + 19T^{2} \)
23 \( 1 - 9.23T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 - 2.76T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 0.472T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 - 6.47T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 + 4.94T + 79T^{2} \)
83 \( 1 - 9.70T + 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 - 7.52T + 97T^{2} \)
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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

−8.826334960644862190620835417062, −7.69795169228579261694256444440, −6.73679711423716793998169247623, −6.47790312580407719777373341497, −5.65816039089081809579845600401, −5.04741161499858237197776803305, −4.11847918061008561489565467081, −3.31505311720982127717586997434, −1.55755587358434184389707832286, −0.72021680117832442543401959750, 0.72021680117832442543401959750, 1.55755587358434184389707832286, 3.31505311720982127717586997434, 4.11847918061008561489565467081, 5.04741161499858237197776803305, 5.65816039089081809579845600401, 6.47790312580407719777373341497, 6.73679711423716793998169247623, 7.69795169228579261694256444440, 8.826334960644862190620835417062

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