Properties

Label 2-1575-1.1-c1-0-30
Degree $2$
Conductor $1575$
Sign $-1$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
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  − 0.618·2-s − 1.61·4-s − 7-s + 2.23·8-s + 0.236·11-s − 1.23·13-s + 0.618·14-s + 1.85·16-s + 2.47·17-s − 4.47·19-s − 0.145·22-s + 6.23·23-s + 0.763·26-s + 1.61·28-s − 5·29-s + 3.70·31-s − 5.61·32-s − 1.52·34-s − 3·37-s + 2.76·38-s − 4.76·41-s − 1.76·43-s − 0.381·44-s − 3.85·46-s − 2·47-s + 49-s + 2.00·52-s + ⋯
L(s)  = 1  − 0.437·2-s − 0.809·4-s − 0.377·7-s + 0.790·8-s + 0.0711·11-s − 0.342·13-s + 0.165·14-s + 0.463·16-s + 0.599·17-s − 1.02·19-s − 0.0311·22-s + 1.30·23-s + 0.149·26-s + 0.305·28-s − 0.928·29-s + 0.666·31-s − 0.993·32-s − 0.262·34-s − 0.493·37-s + 0.448·38-s − 0.744·41-s − 0.268·43-s − 0.0575·44-s − 0.568·46-s − 0.291·47-s + 0.142·49-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 + 0.618T + 2T^{2} \)
11 \( 1 - 0.236T + 11T^{2} \)
13 \( 1 + 1.23T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + 4.47T + 19T^{2} \)
23 \( 1 - 6.23T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 - 3.70T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + 4.76T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 4.23T + 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 8.76T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + 5.23T + 97T^{2} \)
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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

−9.015818767480641590494540269519, −8.414233578723368141617527311173, −7.53148923264926700975358271651, −6.75009883347375055028335954907, −5.66712829094775006233928476008, −4.84122488701842089366980507762, −3.96703907466446787406158731692, −2.93410291197492932117551952451, −1.44631677378036851450710791691, 0, 1.44631677378036851450710791691, 2.93410291197492932117551952451, 3.96703907466446787406158731692, 4.84122488701842089366980507762, 5.66712829094775006233928476008, 6.75009883347375055028335954907, 7.53148923264926700975358271651, 8.414233578723368141617527311173, 9.015818767480641590494540269519

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