Properties

Label 2-3e2-1.1-c25-0-3
Degree $2$
Conductor $9$
Sign $1$
Analytic cond. $35.6397$
Root an. cond. $5.96990$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 48·2-s − 3.35e7·4-s + 7.41e8·5-s + 3.90e10·7-s − 3.22e9·8-s + 3.56e10·10-s − 8.41e12·11-s − 8.16e13·13-s + 1.87e12·14-s + 1.12e15·16-s + 2.51e15·17-s − 6.08e15·19-s − 2.48e16·20-s − 4.04e14·22-s + 9.49e16·23-s + 2.52e17·25-s − 3.91e15·26-s − 1.31e18·28-s + 2.71e17·29-s + 4.29e18·31-s + 1.62e17·32-s + 1.20e17·34-s + 2.89e19·35-s + 2.03e19·37-s − 2.91e17·38-s − 2.39e18·40-s + 1.83e20·41-s + ⋯
L(s)  = 1  + 0.00828·2-s − 0.999·4-s + 1.35·5-s + 1.06·7-s − 0.0165·8-s + 0.0112·10-s − 0.808·11-s − 0.972·13-s + 0.00884·14-s + 0.999·16-s + 1.04·17-s − 0.630·19-s − 1.35·20-s − 0.00670·22-s + 0.903·23-s + 0.847·25-s − 0.00805·26-s − 1.06·28-s + 0.142·29-s + 0.978·31-s + 0.0248·32-s + 0.00869·34-s + 1.45·35-s + 0.506·37-s − 0.00522·38-s − 0.0225·40-s + 1.27·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $1$
Analytic conductor: \(35.6397\)
Root analytic conductor: \(5.96990\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9,\ (\ :25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(2.233920503\)
\(L(\frac12)\) \(\approx\) \(2.233920503\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 3 p^{4} T + p^{25} T^{2} \)
5 \( 1 - 29679594 p^{2} T + p^{25} T^{2} \)
7 \( 1 - 797563208 p^{2} T + p^{25} T^{2} \)
11 \( 1 + 765410481732 p T + p^{25} T^{2} \)
13 \( 1 + 6280849641178 p T + p^{25} T^{2} \)
17 \( 1 - 148229413467534 p T + p^{25} T^{2} \)
19 \( 1 + 320108230016260 p T + p^{25} T^{2} \)
23 \( 1 - 4130229578100888 p T + p^{25} T^{2} \)
29 \( 1 - 271246959476737410 T + p^{25} T^{2} \)
31 \( 1 - 4291666067521509152 T + p^{25} T^{2} \)
37 \( 1 - 20301484446109126982 T + p^{25} T^{2} \)
41 \( 1 - \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} \)
43 \( 1 - \)\(30\!\cdots\!56\)\( T + p^{25} T^{2} \)
47 \( 1 - \)\(92\!\cdots\!88\)\( T + p^{25} T^{2} \)
53 \( 1 - \)\(99\!\cdots\!54\)\( T + p^{25} T^{2} \)
59 \( 1 + \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} \)
61 \( 1 - \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} \)
67 \( 1 + \)\(26\!\cdots\!28\)\( T + p^{25} T^{2} \)
71 \( 1 - \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} \)
73 \( 1 - \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \)
79 \( 1 + \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} \)
83 \( 1 - \)\(93\!\cdots\!84\)\( T + p^{25} T^{2} \)
89 \( 1 - \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} \)
97 \( 1 - \)\(28\!\cdots\!62\)\( T + p^{25} T^{2} \)
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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

−14.73303728071632623015039573762, −13.84221498461638082578393210171, −12.60097969482721422146320854361, −10.48110241612803731034832171616, −9.349959038985898627795854819038, −7.86826029633853092069357759973, −5.63860519233705084872166682308, −4.69879911853203495029984222424, −2.46672246982849191610358557167, −0.959019097659881081229005783175, 0.959019097659881081229005783175, 2.46672246982849191610358557167, 4.69879911853203495029984222424, 5.63860519233705084872166682308, 7.86826029633853092069357759973, 9.349959038985898627795854819038, 10.48110241612803731034832171616, 12.60097969482721422146320854361, 13.84221498461638082578393210171, 14.73303728071632623015039573762

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