Properties

Label 2-75e2-1.1-c1-0-100
Degree $2$
Conductor $5625$
Sign $-1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
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  − 0.541·2-s − 1.70·4-s − 1.08·7-s + 2.00·8-s − 2.91·11-s − 0.966·13-s + 0.589·14-s + 2.32·16-s + 2.65·17-s − 1.96·19-s + 1.57·22-s − 6.83·23-s + 0.523·26-s + 1.85·28-s + 6.81·29-s + 2.48·31-s − 5.27·32-s − 1.44·34-s + 8.22·37-s + 1.06·38-s + 10.5·41-s + 7.72·43-s + 4.97·44-s + 3.70·46-s + 9.02·47-s − 5.81·49-s + 1.64·52-s + ⋯
L(s)  = 1  − 0.383·2-s − 0.853·4-s − 0.411·7-s + 0.710·8-s − 0.878·11-s − 0.267·13-s + 0.157·14-s + 0.581·16-s + 0.645·17-s − 0.451·19-s + 0.336·22-s − 1.42·23-s + 0.102·26-s + 0.350·28-s + 1.26·29-s + 0.446·31-s − 0.932·32-s − 0.247·34-s + 1.35·37-s + 0.172·38-s + 1.65·41-s + 1.17·43-s + 0.749·44-s + 0.546·46-s + 1.31·47-s − 0.830·49-s + 0.228·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 0.541T + 2T^{2} \)
7 \( 1 + 1.08T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 0.966T + 13T^{2} \)
17 \( 1 - 2.65T + 17T^{2} \)
19 \( 1 + 1.96T + 19T^{2} \)
23 \( 1 + 6.83T + 23T^{2} \)
29 \( 1 - 6.81T + 29T^{2} \)
31 \( 1 - 2.48T + 31T^{2} \)
37 \( 1 - 8.22T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 7.72T + 43T^{2} \)
47 \( 1 - 9.02T + 47T^{2} \)
53 \( 1 + 5.53T + 53T^{2} \)
59 \( 1 + 13.2T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 + 8.84T + 67T^{2} \)
71 \( 1 + 7.81T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 6.60T + 79T^{2} \)
83 \( 1 - 1.89T + 83T^{2} \)
89 \( 1 + 1.14T + 89T^{2} \)
97 \( 1 - 6.19T + 97T^{2} \)
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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

−7.72520836940492320999935168519, −7.49144611731552239007876828320, −6.07372258437220583886372470031, −5.85226187201739563214816363794, −4.60368601746362948663273744766, −4.35061276794740853613902616287, −3.19307716242613237605822459817, −2.39106372204962878119590993892, −1.06552298075319220856487162639, 0, 1.06552298075319220856487162639, 2.39106372204962878119590993892, 3.19307716242613237605822459817, 4.35061276794740853613902616287, 4.60368601746362948663273744766, 5.85226187201739563214816363794, 6.07372258437220583886372470031, 7.49144611731552239007876828320, 7.72520836940492320999935168519

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