Properties

Label 2-1075-1.1-c3-0-82
Degree $2$
Conductor $1075$
Sign $-1$
Analytic cond. $63.4270$
Root an. cond. $7.96411$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84·2-s − 9.49·3-s − 4.58·4-s + 17.5·6-s + 26.0·7-s + 23.2·8-s + 63.1·9-s − 36.8·11-s + 43.5·12-s − 89.5·13-s − 48.1·14-s − 6.28·16-s + 28.8·17-s − 116.·18-s − 58.8·19-s − 247.·21-s + 68.0·22-s − 2.63·23-s − 220.·24-s + 165.·26-s − 343.·27-s − 119.·28-s + 173.·29-s + 57.9·31-s − 174.·32-s + 349.·33-s − 53.2·34-s + ⋯
L(s)  = 1  − 0.653·2-s − 1.82·3-s − 0.573·4-s + 1.19·6-s + 1.40·7-s + 1.02·8-s + 2.34·9-s − 1.01·11-s + 1.04·12-s − 1.91·13-s − 0.919·14-s − 0.0981·16-s + 0.410·17-s − 1.52·18-s − 0.710·19-s − 2.57·21-s + 0.659·22-s − 0.0238·23-s − 1.87·24-s + 1.24·26-s − 2.44·27-s − 0.806·28-s + 1.11·29-s + 0.335·31-s − 0.963·32-s + 1.84·33-s − 0.268·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(63.4270\)
Root analytic conductor: \(7.96411\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1075,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - 43T \)
good2 \( 1 + 1.84T + 8T^{2} \)
3 \( 1 + 9.49T + 27T^{2} \)
7 \( 1 - 26.0T + 343T^{2} \)
11 \( 1 + 36.8T + 1.33e3T^{2} \)
13 \( 1 + 89.5T + 2.19e3T^{2} \)
17 \( 1 - 28.8T + 4.91e3T^{2} \)
19 \( 1 + 58.8T + 6.85e3T^{2} \)
23 \( 1 + 2.63T + 1.21e4T^{2} \)
29 \( 1 - 173.T + 2.43e4T^{2} \)
31 \( 1 - 57.9T + 2.97e4T^{2} \)
37 \( 1 + 52.0T + 5.06e4T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
47 \( 1 - 106.T + 1.03e5T^{2} \)
53 \( 1 + 244.T + 1.48e5T^{2} \)
59 \( 1 + 127.T + 2.05e5T^{2} \)
61 \( 1 + 443.T + 2.26e5T^{2} \)
67 \( 1 - 117.T + 3.00e5T^{2} \)
71 \( 1 - 816.T + 3.57e5T^{2} \)
73 \( 1 - 620.T + 3.89e5T^{2} \)
79 \( 1 - 377.T + 4.93e5T^{2} \)
83 \( 1 - 1.45e3T + 5.71e5T^{2} \)
89 \( 1 - 627.T + 7.04e5T^{2} \)
97 \( 1 - 817.T + 9.12e5T^{2} \)
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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

−9.299289422882928470050996751533, −7.939231856574242179755173056794, −7.68042777299781025478420300442, −6.58622762389336525158065044884, −5.28310341169522509238356423029, −4.97655815452506450970013421097, −4.37601689366677805999786377490, −2.14069552771751241142050005613, −0.915823438129458757326369100566, 0, 0.915823438129458757326369100566, 2.14069552771751241142050005613, 4.37601689366677805999786377490, 4.97655815452506450970013421097, 5.28310341169522509238356423029, 6.58622762389336525158065044884, 7.68042777299781025478420300442, 7.939231856574242179755173056794, 9.299289422882928470050996751533

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