Properties

Label 2-245-1.1-c1-0-11
Degree $2$
Conductor $245$
Sign $-1$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 5-s − 2·9-s − 3·11-s + 2·12-s − 5·13-s − 15-s + 4·16-s − 3·17-s − 2·19-s − 2·20-s − 6·23-s + 25-s + 5·27-s + 3·29-s + 4·31-s + 3·33-s + 4·36-s + 2·37-s + 5·39-s + 12·41-s − 10·43-s + 6·44-s − 2·45-s − 9·47-s − 4·48-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 1.38·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.458·19-s − 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.522·33-s + 2/3·36-s + 0.328·37-s + 0.800·39-s + 1.87·41-s − 1.52·43-s + 0.904·44-s − 0.298·45-s − 1.31·47-s − 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−11.72263911909931664995468447198, −10.46766423688804548750268495569, −9.833804192038460581265801340645, −8.722193961422815471879095064652, −7.80156257853413862136981277007, −6.30078072165756181986797640838, −5.30121167828360660438204787375, −4.48390865811726709341858080699, −2.61198254943438177491615875407, 0, 2.61198254943438177491615875407, 4.48390865811726709341858080699, 5.30121167828360660438204787375, 6.30078072165756181986797640838, 7.80156257853413862136981277007, 8.722193961422815471879095064652, 9.833804192038460581265801340645, 10.46766423688804548750268495569, 11.72263911909931664995468447198

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