Properties

Label 2-1575-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s − 1.82·4-s + 7-s + 1.58·8-s − 4.82·11-s − 0.828·13-s − 0.414·14-s + 3·16-s + 7.65·17-s − 2.82·19-s + 1.99·22-s − 3.65·23-s + 0.343·26-s − 1.82·28-s − 8·29-s + 8.48·31-s − 4.41·32-s − 3.17·34-s + 6·37-s + 1.17·38-s + 7.65·41-s − 1.65·43-s + 8.82·44-s + 1.51·46-s − 4·47-s + 49-s + 1.51·52-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + 0.377·7-s + 0.560·8-s − 1.45·11-s − 0.229·13-s − 0.110·14-s + 0.750·16-s + 1.85·17-s − 0.648·19-s + 0.426·22-s − 0.762·23-s + 0.0672·26-s − 0.345·28-s − 1.48·29-s + 1.52·31-s − 0.780·32-s − 0.543·34-s + 0.986·37-s + 0.190·38-s + 1.19·41-s − 0.252·43-s + 1.33·44-s + 0.223·46-s − 0.583·47-s + 0.142·49-s + 0.210·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019626805\)
\(L(\frac12)\) \(\approx\) \(1.019626805\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 15.3T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 - 5.31T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
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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

−9.640639981803127140396868550522, −8.417711186473193927394013829548, −7.982400768327329245737691485838, −7.40817959548833171568646731378, −5.91750772154343333472196384105, −5.30865918421852678820916636976, −4.48312969709789838914568224774, −3.48643949906143382505202783646, −2.25668407264442344072386569946, −0.74640220026103708259968382766, 0.74640220026103708259968382766, 2.25668407264442344072386569946, 3.48643949906143382505202783646, 4.48312969709789838914568224774, 5.30865918421852678820916636976, 5.91750772154343333472196384105, 7.40817959548833171568646731378, 7.982400768327329245737691485838, 8.417711186473193927394013829548, 9.640639981803127140396868550522

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