Properties

Label 2-4275-1.1-c1-0-10
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  − 0.414·2-s − 1.82·4-s − 1.41·7-s + 1.58·8-s − 6.24·11-s + 0.585·13-s + 0.585·14-s + 3·16-s + 6.82·17-s − 19-s + 2.58·22-s − 3.65·23-s − 0.242·26-s + 2.58·28-s + 1.41·29-s − 8.82·31-s − 4.41·32-s − 2.82·34-s + 0.585·37-s + 0.414·38-s − 8.24·41-s − 3.75·43-s + 11.4·44-s + 1.51·46-s + 3.65·47-s − 5·49-s − 1.07·52-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s − 0.534·7-s + 0.560·8-s − 1.88·11-s + 0.162·13-s + 0.156·14-s + 0.750·16-s + 1.65·17-s − 0.229·19-s + 0.551·22-s − 0.762·23-s − 0.0475·26-s + 0.488·28-s + 0.262·29-s − 1.58·31-s − 0.780·32-s − 0.485·34-s + 0.0963·37-s + 0.0671·38-s − 1.28·41-s − 0.572·43-s + 1.72·44-s + 0.223·46-s + 0.533·47-s − 0.714·49-s − 0.148·52-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: $\chi_{4275} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4275,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6653533104\)
\(L(\frac12)\) \(\approx\) \(0.6653533104\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 + 6.24T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 - 0.585T + 37T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 - 3.65T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 - 4.48T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 1.65T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 7.17T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−8.264103139920201003462964828883, −7.84648688006282962233995783039, −7.19554373802638507657587415396, −5.96891106224992177388979528260, −5.40624452326261759498993887652, −4.79326861007076927693510536819, −3.67669524627480762254298259072, −3.09777096013286524038510214081, −1.85819906304243172351811330659, −0.47413381271937967808859363874, 0.47413381271937967808859363874, 1.85819906304243172351811330659, 3.09777096013286524038510214081, 3.67669524627480762254298259072, 4.79326861007076927693510536819, 5.40624452326261759498993887652, 5.96891106224992177388979528260, 7.19554373802638507657587415396, 7.84648688006282962233995783039, 8.264103139920201003462964828883

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