Properties

Label 2-445-1.1-c1-0-14
Degree $2$
Conductor $445$
Sign $-1$
Analytic cond. $3.55334$
Root an. cond. $1.88503$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.75·2-s − 0.459·3-s + 5.56·4-s + 5-s + 1.26·6-s + 0.587·7-s − 9.81·8-s − 2.78·9-s − 2.75·10-s − 0.892·11-s − 2.56·12-s − 1.31·13-s − 1.61·14-s − 0.459·15-s + 15.8·16-s − 6.89·17-s + 7.67·18-s + 2.78·19-s + 5.56·20-s − 0.270·21-s + 2.45·22-s − 0.636·23-s + 4.51·24-s + 25-s + 3.61·26-s + 2.66·27-s + 3.27·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.265·3-s + 2.78·4-s + 0.447·5-s + 0.516·6-s + 0.222·7-s − 3.47·8-s − 0.929·9-s − 0.869·10-s − 0.268·11-s − 0.739·12-s − 0.364·13-s − 0.432·14-s − 0.118·15-s + 3.96·16-s − 1.67·17-s + 1.80·18-s + 0.639·19-s + 1.24·20-s − 0.0589·21-s + 0.523·22-s − 0.132·23-s + 0.921·24-s + 0.200·25-s + 0.709·26-s + 0.512·27-s + 0.618·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(445\)    =    \(5 \cdot 89\)
Sign: $-1$
Analytic conductor: \(3.55334\)
Root analytic conductor: \(1.88503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 445,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
89 \( 1 - T \)
good2 \( 1 + 2.75T + 2T^{2} \)
3 \( 1 + 0.459T + 3T^{2} \)
7 \( 1 - 0.587T + 7T^{2} \)
11 \( 1 + 0.892T + 11T^{2} \)
13 \( 1 + 1.31T + 13T^{2} \)
17 \( 1 + 6.89T + 17T^{2} \)
19 \( 1 - 2.78T + 19T^{2} \)
23 \( 1 + 0.636T + 23T^{2} \)
29 \( 1 - 5.55T + 29T^{2} \)
31 \( 1 + 9.03T + 31T^{2} \)
37 \( 1 - 0.632T + 37T^{2} \)
41 \( 1 + 7.36T + 41T^{2} \)
43 \( 1 + 7.41T + 43T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 - 3.18T + 53T^{2} \)
59 \( 1 - 5.68T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 - 7.78T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−10.50135114093515880110584856330, −9.692542928512956046648912166420, −8.815249615773039709526197231051, −8.256213937658395535298854455860, −7.10421504866325048064708776598, −6.36944103482320413140153797868, −5.25319731601321216044861769518, −2.93780312850244441451367290352, −1.83296779380478863816478260892, 0, 1.83296779380478863816478260892, 2.93780312850244441451367290352, 5.25319731601321216044861769518, 6.36944103482320413140153797868, 7.10421504866325048064708776598, 8.256213937658395535298854455860, 8.815249615773039709526197231051, 9.692542928512956046648912166420, 10.50135114093515880110584856330

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