Properties

Label 2-75e2-1.1-c1-0-108
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  − 1.61·2-s + 0.618·4-s − 2·7-s + 2.23·8-s + 3·11-s − 13-s + 3.23·14-s − 4.85·16-s + 4.23·17-s − 6.70·19-s − 4.85·22-s − 5.38·23-s + 1.61·26-s − 1.23·28-s + 3.61·29-s + 8.70·31-s + 3.38·32-s − 6.85·34-s − 2·37-s + 10.8·38-s + 9.38·41-s − 7.38·43-s + 1.85·44-s + 8.70·46-s + 4.76·47-s − 3·49-s − 0.618·52-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s − 0.755·7-s + 0.790·8-s + 0.904·11-s − 0.277·13-s + 0.864·14-s − 1.21·16-s + 1.02·17-s − 1.53·19-s − 1.03·22-s − 1.12·23-s + 0.317·26-s − 0.233·28-s + 0.671·29-s + 1.56·31-s + 0.597·32-s − 1.17·34-s − 0.328·37-s + 1.76·38-s + 1.46·41-s − 1.12·43-s + 0.279·44-s + 1.28·46-s + 0.694·47-s − 0.428·49-s − 0.0857·52-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: $\chi_{5625} (1, \cdot )$
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 \( 1 + 1.61T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 4.23T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 + 5.38T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 9.38T + 41T^{2} \)
43 \( 1 + 7.38T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 + 9.14T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 3.85T + 97T^{2} \)
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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

−8.058761387913769390759206754715, −7.18944872473669821507225983523, −6.45184662222692698774381120346, −5.97871628363432092026064896772, −4.68400757169133059369707768368, −4.14178004347461465395548155382, −3.14195708488296106043979789671, −2.08136400647833211258189485686, −1.09820080009033362205797219087, 0, 1.09820080009033362205797219087, 2.08136400647833211258189485686, 3.14195708488296106043979789671, 4.14178004347461465395548155382, 4.68400757169133059369707768368, 5.97871628363432092026064896772, 6.45184662222692698774381120346, 7.18944872473669821507225983523, 8.058761387913769390759206754715

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