Properties

Label 2-65e2-1.1-c1-0-96
Degree $2$
Conductor $4225$
Sign $-1$
Analytic cond. $33.7367$
Root an. cond. $5.80833$
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  − 2.30·2-s − 3-s + 3.30·4-s + 2.30·6-s − 7-s − 3.00·8-s − 2·9-s − 1.60·11-s − 3.30·12-s + 2.30·14-s + 0.302·16-s − 7.60·17-s + 4.60·18-s + 5.60·19-s + 21-s + 3.69·22-s + 3·23-s + 3.00·24-s + 5·27-s − 3.30·28-s − 6.21·29-s + 4·31-s + 5.30·32-s + 1.60·33-s + 17.5·34-s − 6.60·36-s + 3.60·37-s + ⋯
L(s)  = 1  − 1.62·2-s − 0.577·3-s + 1.65·4-s + 0.940·6-s − 0.377·7-s − 1.06·8-s − 0.666·9-s − 0.484·11-s − 0.953·12-s + 0.615·14-s + 0.0756·16-s − 1.84·17-s + 1.08·18-s + 1.28·19-s + 0.218·21-s + 0.788·22-s + 0.625·23-s + 0.612·24-s + 0.962·27-s − 0.624·28-s − 1.15·29-s + 0.718·31-s + 0.937·32-s + 0.279·33-s + 3.00·34-s − 1.10·36-s + 0.592·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4225\)    =    \(5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(33.7367\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4225,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + 2.30T + 2T^{2} \)
3 \( 1 + T + 3T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 1.60T + 11T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + 6.21T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 3.60T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 - 3.21T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 4.81T + 71T^{2} \)
73 \( 1 + 0.788T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 - 6.21T + 89T^{2} \)
97 \( 1 + 8.39T + 97T^{2} \)
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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

−8.138974301942529904026999769557, −7.39452526759548797405133795775, −6.81812031924390525880533038960, −6.05317710539572665001307904888, −5.28840309853066241515613712858, −4.29404482096101849856557035076, −2.94158310033743530616353188925, −2.25334640849633247991995982842, −0.936578016626246354578999037112, 0, 0.936578016626246354578999037112, 2.25334640849633247991995982842, 2.94158310033743530616353188925, 4.29404482096101849856557035076, 5.28840309853066241515613712858, 6.05317710539572665001307904888, 6.81812031924390525880533038960, 7.39452526759548797405133795775, 8.138974301942529904026999769557

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