Properties

Label 2-75e2-1.1-c1-0-169
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  + 1.23·2-s − 0.468·4-s + 2.08·7-s − 3.05·8-s + 2.65·11-s − 0.149·13-s + 2.58·14-s − 2.84·16-s − 5.12·17-s − 3.08·19-s + 3.28·22-s + 2.76·23-s − 0.185·26-s − 0.977·28-s − 6.94·29-s + 1.52·31-s + 2.58·32-s − 6.34·34-s − 8.52·37-s − 3.81·38-s + 9.82·41-s − 3.43·43-s − 1.24·44-s + 3.41·46-s − 4.00·47-s − 2.64·49-s + 0.0701·52-s + ⋯
L(s)  = 1  + 0.875·2-s − 0.234·4-s + 0.788·7-s − 1.08·8-s + 0.799·11-s − 0.0415·13-s + 0.690·14-s − 0.711·16-s − 1.24·17-s − 0.708·19-s + 0.699·22-s + 0.576·23-s − 0.0363·26-s − 0.184·28-s − 1.28·29-s + 0.273·31-s + 0.457·32-s − 1.08·34-s − 1.40·37-s − 0.619·38-s + 1.53·41-s − 0.523·43-s − 0.187·44-s + 0.504·46-s − 0.584·47-s − 0.378·49-s + 0.00972·52-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 \( 1 - 1.23T + 2T^{2} \)
7 \( 1 - 2.08T + 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + 0.149T + 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
23 \( 1 - 2.76T + 23T^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + 8.52T + 37T^{2} \)
41 \( 1 - 9.82T + 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 4.00T + 47T^{2} \)
53 \( 1 + 0.945T + 53T^{2} \)
59 \( 1 + 8.18T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 3.34T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 3.45T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 0.0571T + 97T^{2} \)
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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

−7.79628331616691019482775225967, −6.81510818063796143844291971385, −6.31219037791308342799211329766, −5.43237089878884271506605922419, −4.77248695725208160058739317306, −4.19943115846393173555657169634, −3.52452572625373174301221815875, −2.46530884317195469737717973359, −1.51316137125109064892158598114, 0, 1.51316137125109064892158598114, 2.46530884317195469737717973359, 3.52452572625373174301221815875, 4.19943115846393173555657169634, 4.77248695725208160058739317306, 5.43237089878884271506605922419, 6.31219037791308342799211329766, 6.81510818063796143844291971385, 7.79628331616691019482775225967

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