Properties

Label 2-145-1.1-c1-0-1
Degree $2$
Conductor $145$
Sign $1$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  − 1.48·2-s + 0.806·3-s + 0.193·4-s + 5-s − 1.19·6-s + 1.19·7-s + 2.67·8-s − 2.35·9-s − 1.48·10-s + 4.15·11-s + 0.156·12-s + 2.96·13-s − 1.76·14-s + 0.806·15-s − 4.35·16-s + 5.50·17-s + 3.48·18-s − 3.19·19-s + 0.193·20-s + 0.962·21-s − 6.15·22-s + 1.84·23-s + 2.15·24-s + 25-s − 4.38·26-s − 4.31·27-s + 0.231·28-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.465·3-s + 0.0969·4-s + 0.447·5-s − 0.487·6-s + 0.451·7-s + 0.945·8-s − 0.783·9-s − 0.468·10-s + 1.25·11-s + 0.0451·12-s + 0.821·13-s − 0.472·14-s + 0.208·15-s − 1.08·16-s + 1.33·17-s + 0.820·18-s − 0.732·19-s + 0.0433·20-s + 0.210·21-s − 1.31·22-s + 0.384·23-s + 0.440·24-s + 0.200·25-s − 0.860·26-s − 0.829·27-s + 0.0437·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $1$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8172581684\)
\(L(\frac12)\) \(\approx\) \(0.8172581684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + 1.48T + 2T^{2} \)
3 \( 1 - 0.806T + 3T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 - 2.96T + 13T^{2} \)
17 \( 1 - 5.50T + 17T^{2} \)
19 \( 1 + 3.19T + 19T^{2} \)
23 \( 1 - 1.84T + 23T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 + 9.50T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 0.0303T + 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 + 1.35T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 - 5.84T + 67T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 4.93T + 79T^{2} \)
83 \( 1 - 4.41T + 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 + 1.38T + 97T^{2} \)
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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

−13.27812782785293292925916687227, −11.83621085492559691451843086283, −10.82866461448374823144662632684, −9.775325787434878633437529873691, −8.797368904858264316058028308175, −8.321737527789852573393432076807, −6.92649464536852554055735097192, −5.44699759480368652377904833707, −3.69449252151016401999673513400, −1.58907978642613073609715688098, 1.58907978642613073609715688098, 3.69449252151016401999673513400, 5.44699759480368652377904833707, 6.92649464536852554055735097192, 8.321737527789852573393432076807, 8.797368904858264316058028308175, 9.775325787434878633437529873691, 10.82866461448374823144662632684, 11.83621085492559691451843086283, 13.27812782785293292925916687227

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