Properties

Label 2-4005-1.1-c1-0-34
Degree $2$
Conductor $4005$
Sign $1$
Analytic cond. $31.9800$
Root an. cond. $5.65509$
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  − 2.74·2-s + 5.55·4-s − 5-s + 4.26·7-s − 9.77·8-s + 2.74·10-s + 0.899·11-s − 6.86·13-s − 11.7·14-s + 15.7·16-s + 2.01·17-s + 8.38·19-s − 5.55·20-s − 2.47·22-s + 6.70·23-s + 25-s + 18.8·26-s + 23.6·28-s − 2.96·29-s − 5.52·31-s − 23.7·32-s − 5.53·34-s − 4.26·35-s − 5.73·37-s − 23.0·38-s + 9.77·40-s + 1.02·41-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.77·4-s − 0.447·5-s + 1.61·7-s − 3.45·8-s + 0.869·10-s + 0.271·11-s − 1.90·13-s − 3.13·14-s + 3.93·16-s + 0.488·17-s + 1.92·19-s − 1.24·20-s − 0.527·22-s + 1.39·23-s + 0.200·25-s + 3.70·26-s + 4.47·28-s − 0.550·29-s − 0.992·31-s − 4.20·32-s − 0.949·34-s − 0.720·35-s − 0.942·37-s − 3.73·38-s + 1.54·40-s + 0.160·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $1$
Analytic conductor: \(31.9800\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4005,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8509463623\)
\(L(\frac12)\) \(\approx\) \(0.8509463623\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
89 \( 1 - T \)
good2 \( 1 + 2.74T + 2T^{2} \)
7 \( 1 - 4.26T + 7T^{2} \)
11 \( 1 - 0.899T + 11T^{2} \)
13 \( 1 + 6.86T + 13T^{2} \)
17 \( 1 - 2.01T + 17T^{2} \)
19 \( 1 - 8.38T + 19T^{2} \)
23 \( 1 - 6.70T + 23T^{2} \)
29 \( 1 + 2.96T + 29T^{2} \)
31 \( 1 + 5.52T + 31T^{2} \)
37 \( 1 + 5.73T + 37T^{2} \)
41 \( 1 - 1.02T + 41T^{2} \)
43 \( 1 - 5.77T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 - 5.88T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 - 4.46T + 61T^{2} \)
67 \( 1 + 4.31T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 4.95T + 73T^{2} \)
79 \( 1 + 7.52T + 79T^{2} \)
83 \( 1 - 0.785T + 83T^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−8.432884677810245840503942134660, −7.64703313026830730102082571832, −7.43701902493229751060275501653, −6.87575481328484712733785925825, −5.39010991232001918463784248512, −5.04811619707674044625145184789, −3.47752696164341906922026595969, −2.50147253696206395770712899162, −1.62182776056323557368606920765, −0.74009063415119550128182337612, 0.74009063415119550128182337612, 1.62182776056323557368606920765, 2.50147253696206395770712899162, 3.47752696164341906922026595969, 5.04811619707674044625145184789, 5.39010991232001918463784248512, 6.87575481328484712733785925825, 7.43701902493229751060275501653, 7.64703313026830730102082571832, 8.432884677810245840503942134660

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