Properties

Label 2-4275-1.1-c1-0-80
Degree $2$
Conductor $4275$
Sign $1$
Analytic cond. $34.1360$
Root an. cond. $5.84260$
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  + 2.37·2-s + 3.65·4-s − 0.726·7-s + 3.92·8-s + 0.273·11-s + 5.95·13-s − 1.72·14-s + 2.02·16-s + 5.27·17-s + 19-s + 0.651·22-s − 3.67·23-s + 14.1·26-s − 2.65·28-s + 2.27·29-s + 3.19·31-s − 3.02·32-s + 12.5·34-s − 8.12·37-s + 2.37·38-s + 9.43·41-s − 9.81·43-s + 44-s − 8.74·46-s + 12.1·47-s − 6.47·49-s + 21.7·52-s + ⋯
L(s)  = 1  + 1.68·2-s + 1.82·4-s − 0.274·7-s + 1.38·8-s + 0.0825·11-s + 1.65·13-s − 0.461·14-s + 0.507·16-s + 1.27·17-s + 0.229·19-s + 0.138·22-s − 0.767·23-s + 2.77·26-s − 0.501·28-s + 0.422·29-s + 0.574·31-s − 0.535·32-s + 2.15·34-s − 1.33·37-s + 0.385·38-s + 1.47·41-s − 1.49·43-s + 0.150·44-s − 1.28·46-s + 1.77·47-s − 0.924·49-s + 3.01·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4275\)    =    \(3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(34.1360\)
Root analytic conductor: \(5.84260\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4275,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.866890779\)
\(L(\frac12)\) \(\approx\) \(5.866890779\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good2 \( 1 - 2.37T + 2T^{2} \)
7 \( 1 + 0.726T + 7T^{2} \)
11 \( 1 - 0.273T + 11T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
23 \( 1 + 3.67T + 23T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 8.12T + 37T^{2} \)
41 \( 1 - 9.43T + 41T^{2} \)
43 \( 1 + 9.81T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 + 0.103T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 5.75T + 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 - 3.87T + 79T^{2} \)
83 \( 1 + 0.488T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 4.44T + 97T^{2} \)
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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

−8.260463990297788191630253105754, −7.40431745638240561064337112510, −6.57783274606217489612406022962, −5.98327309663853387818180422712, −5.48402081525655015474837100765, −4.59537704372274093724485294848, −3.66535353405405435315061388968, −3.40993454429543777336460815027, −2.31121733241419916104656973415, −1.15065948936924680686550817808, 1.15065948936924680686550817808, 2.31121733241419916104656973415, 3.40993454429543777336460815027, 3.66535353405405435315061388968, 4.59537704372274093724485294848, 5.48402081525655015474837100765, 5.98327309663853387818180422712, 6.57783274606217489612406022962, 7.40431745638240561064337112510, 8.260463990297788191630253105754

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