Properties

Label 2-4025-1.1-c1-0-87
Degree $2$
Conductor $4025$
Sign $1$
Analytic cond. $32.1397$
Root an. cond. $5.66919$
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  + 0.556·2-s + 2.94·3-s − 1.69·4-s + 1.63·6-s − 7-s − 2.05·8-s + 5.65·9-s − 5.83·11-s − 4.97·12-s + 4.53·13-s − 0.556·14-s + 2.24·16-s + 3.00·17-s + 3.14·18-s + 3.78·19-s − 2.94·21-s − 3.24·22-s − 23-s − 6.03·24-s + 2.52·26-s + 7.80·27-s + 1.69·28-s − 6.57·29-s + 6.03·31-s + 5.35·32-s − 17.1·33-s + 1.66·34-s + ⋯
L(s)  = 1  + 0.393·2-s + 1.69·3-s − 0.845·4-s + 0.667·6-s − 0.377·7-s − 0.725·8-s + 1.88·9-s − 1.75·11-s − 1.43·12-s + 1.25·13-s − 0.148·14-s + 0.560·16-s + 0.727·17-s + 0.741·18-s + 0.867·19-s − 0.641·21-s − 0.691·22-s − 0.208·23-s − 1.23·24-s + 0.494·26-s + 1.50·27-s + 0.319·28-s − 1.22·29-s + 1.08·31-s + 0.945·32-s − 2.98·33-s + 0.286·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(32.1397\)
Root analytic conductor: \(5.66919\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.339782062\)
\(L(\frac12)\) \(\approx\) \(3.339782062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good2 \( 1 - 0.556T + 2T^{2} \)
3 \( 1 - 2.94T + 3T^{2} \)
11 \( 1 + 5.83T + 11T^{2} \)
13 \( 1 - 4.53T + 13T^{2} \)
17 \( 1 - 3.00T + 17T^{2} \)
19 \( 1 - 3.78T + 19T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 - 6.03T + 31T^{2} \)
37 \( 1 - 2.03T + 37T^{2} \)
41 \( 1 - 8.22T + 41T^{2} \)
43 \( 1 - 8.07T + 43T^{2} \)
47 \( 1 - 8.34T + 47T^{2} \)
53 \( 1 - 5.57T + 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 1.32T + 61T^{2} \)
67 \( 1 + 0.934T + 67T^{2} \)
71 \( 1 + 0.463T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 5.34T + 79T^{2} \)
83 \( 1 - 9.37T + 83T^{2} \)
89 \( 1 + 15.5T + 89T^{2} \)
97 \( 1 - 3.55T + 97T^{2} \)
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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

−8.368419072349861593715026207237, −7.86552930907815291687929500099, −7.40584784638680826446252242318, −6.00902984088498337262372443112, −5.44551531748727249918709582504, −4.40240692935951100666326187540, −3.68534627748355986454365424458, −3.07262031835963364585247190515, −2.39890561095783072446320816373, −0.933831532304327378021629065603, 0.933831532304327378021629065603, 2.39890561095783072446320816373, 3.07262031835963364585247190515, 3.68534627748355986454365424458, 4.40240692935951100666326187540, 5.44551531748727249918709582504, 6.00902984088498337262372443112, 7.40584784638680826446252242318, 7.86552930907815291687929500099, 8.368419072349861593715026207237

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