Properties

Label 2-75e2-1.1-c1-0-154
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64·2-s + 4.97·4-s + 4.34·7-s + 7.84·8-s − 2.53·11-s − 2.22·13-s + 11.4·14-s + 10.7·16-s + 4.51·17-s + 0.864·19-s − 6.69·22-s + 0.357·23-s − 5.88·26-s + 21.6·28-s + 4.40·29-s − 0.455·31-s + 12.7·32-s + 11.9·34-s − 2.37·37-s + 2.28·38-s − 2.77·41-s − 10.2·43-s − 12.5·44-s + 0.943·46-s + 7.92·47-s + 11.8·49-s − 11.0·52-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.48·4-s + 1.64·7-s + 2.77·8-s − 0.763·11-s − 0.617·13-s + 3.06·14-s + 2.69·16-s + 1.09·17-s + 0.198·19-s − 1.42·22-s + 0.0744·23-s − 1.15·26-s + 4.08·28-s + 0.818·29-s − 0.0817·31-s + 2.25·32-s + 2.04·34-s − 0.390·37-s + 0.370·38-s − 0.433·41-s − 1.56·43-s − 1.89·44-s + 0.139·46-s + 1.15·47-s + 1.69·49-s − 1.53·52-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: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.374192229\)
\(L(\frac12)\) \(\approx\) \(8.374192229\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 2T^{2} \)
7 \( 1 - 4.34T + 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
13 \( 1 + 2.22T + 13T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 - 0.864T + 19T^{2} \)
23 \( 1 - 0.357T + 23T^{2} \)
29 \( 1 - 4.40T + 29T^{2} \)
31 \( 1 + 0.455T + 31T^{2} \)
37 \( 1 + 2.37T + 37T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 7.92T + 47T^{2} \)
53 \( 1 - 6.45T + 53T^{2} \)
59 \( 1 + 6.91T + 59T^{2} \)
61 \( 1 + 6.10T + 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 + 7.37T + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 0.336T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 7.78T + 97T^{2} \)
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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

−7.86742667130263180381616724875, −7.31560433443087578808416305926, −6.55474770396807210161698065797, −5.49192235190019900551271805494, −5.22034959056203719781573710142, −4.66526476524665561362306954516, −3.84965889654426245771507020745, −2.95004426726756107233770584875, −2.20644566667632845351784118471, −1.32445361674399919374106674092, 1.32445361674399919374106674092, 2.20644566667632845351784118471, 2.95004426726756107233770584875, 3.84965889654426245771507020745, 4.66526476524665561362306954516, 5.22034959056203719781573710142, 5.49192235190019900551271805494, 6.55474770396807210161698065797, 7.31560433443087578808416305926, 7.86742667130263180381616724875

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