Properties

Label 2-75e2-1.1-c1-0-162
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  + 0.326·2-s − 1.89·4-s + 3.42·7-s − 1.27·8-s + 5.34·11-s − 3.52·13-s + 1.11·14-s + 3.37·16-s − 2.55·17-s − 2.02·19-s + 1.74·22-s − 7.57·23-s − 1.15·26-s − 6.48·28-s − 4.74·29-s + 1.62·31-s + 3.64·32-s − 0.835·34-s − 0.0134·37-s − 0.661·38-s − 9.67·41-s + 2.32·43-s − 10.1·44-s − 2.47·46-s − 6.94·47-s + 4.72·49-s + 6.66·52-s + ⋯
L(s)  = 1  + 0.231·2-s − 0.946·4-s + 1.29·7-s − 0.449·8-s + 1.61·11-s − 0.976·13-s + 0.299·14-s + 0.842·16-s − 0.620·17-s − 0.464·19-s + 0.372·22-s − 1.57·23-s − 0.225·26-s − 1.22·28-s − 0.880·29-s + 0.291·31-s + 0.644·32-s − 0.143·34-s − 0.00220·37-s − 0.107·38-s − 1.51·41-s + 0.354·43-s − 1.52·44-s − 0.364·46-s − 1.01·47-s + 0.674·49-s + 0.924·52-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 \( 1 - 0.326T + 2T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
11 \( 1 - 5.34T + 11T^{2} \)
13 \( 1 + 3.52T + 13T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 + 2.02T + 19T^{2} \)
23 \( 1 + 7.57T + 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 - 1.62T + 31T^{2} \)
37 \( 1 + 0.0134T + 37T^{2} \)
41 \( 1 + 9.67T + 41T^{2} \)
43 \( 1 - 2.32T + 43T^{2} \)
47 \( 1 + 6.94T + 47T^{2} \)
53 \( 1 + 1.72T + 53T^{2} \)
59 \( 1 - 0.0221T + 59T^{2} \)
61 \( 1 - 3.91T + 61T^{2} \)
67 \( 1 - 4.11T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 - 1.51T + 73T^{2} \)
79 \( 1 - 0.426T + 79T^{2} \)
83 \( 1 - 6.04T + 83T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−8.079076324302768645659927312737, −7.03924698409701099726082012019, −6.31262397793307371458183322321, −5.44876118426433355996237074982, −4.73621062207105718800269714658, −4.20918179062111219344780906730, −3.58006535812577732646540289733, −2.18332518224322208102246193960, −1.41486217621326607143574485847, 0, 1.41486217621326607143574485847, 2.18332518224322208102246193960, 3.58006535812577732646540289733, 4.20918179062111219344780906730, 4.73621062207105718800269714658, 5.44876118426433355996237074982, 6.31262397793307371458183322321, 7.03924698409701099726082012019, 8.079076324302768645659927312737

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