Properties

Label 2-4025-1.1-c1-0-167
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.38·2-s + 1.77·3-s − 0.0920·4-s − 2.44·6-s + 7-s + 2.88·8-s + 0.141·9-s − 0.183·11-s − 0.163·12-s + 1.86·13-s − 1.38·14-s − 3.80·16-s − 0.498·17-s − 0.194·18-s − 1.56·19-s + 1.77·21-s + 0.253·22-s + 23-s + 5.12·24-s − 2.57·26-s − 5.06·27-s − 0.0920·28-s − 7.91·29-s − 1.77·31-s − 0.520·32-s − 0.324·33-s + 0.688·34-s + ⋯
L(s)  = 1  − 0.976·2-s + 1.02·3-s − 0.0460·4-s − 0.999·6-s + 0.377·7-s + 1.02·8-s + 0.0470·9-s − 0.0552·11-s − 0.0471·12-s + 0.516·13-s − 0.369·14-s − 0.951·16-s − 0.120·17-s − 0.0459·18-s − 0.358·19-s + 0.386·21-s + 0.0539·22-s + 0.208·23-s + 1.04·24-s − 0.504·26-s − 0.975·27-s − 0.0173·28-s − 1.46·29-s − 0.318·31-s − 0.0919·32-s − 0.0565·33-s + 0.118·34-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: \(1\)
Selberg data: \((2,\ 4025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
23 \( 1 - T \)
good2 \( 1 + 1.38T + 2T^{2} \)
3 \( 1 - 1.77T + 3T^{2} \)
11 \( 1 + 0.183T + 11T^{2} \)
13 \( 1 - 1.86T + 13T^{2} \)
17 \( 1 + 0.498T + 17T^{2} \)
19 \( 1 + 1.56T + 19T^{2} \)
29 \( 1 + 7.91T + 29T^{2} \)
31 \( 1 + 1.77T + 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 + 8.71T + 41T^{2} \)
43 \( 1 + 1.97T + 43T^{2} \)
47 \( 1 - 5.02T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 5.67T + 59T^{2} \)
61 \( 1 - 8.13T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 + 13.7T + 73T^{2} \)
79 \( 1 - 9.94T + 79T^{2} \)
83 \( 1 + 5.39T + 83T^{2} \)
89 \( 1 - 7.63T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.379472181149085533961888944786, −7.54548094422564379229473409458, −7.09412124005657484978796570866, −5.87779528788240935771370327665, −5.04406806656527569355864930739, −4.05619372816312579362408381056, −3.37918624904589597934577407833, −2.20381889153317641860109514603, −1.48196137297642158962142068057, 0, 1.48196137297642158962142068057, 2.20381889153317641860109514603, 3.37918624904589597934577407833, 4.05619372816312579362408381056, 5.04406806656527569355864930739, 5.87779528788240935771370327665, 7.09412124005657484978796570866, 7.54548094422564379229473409458, 8.379472181149085533961888944786

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