Properties

Label 2-75348-1.1-c1-0-8
Degree $2$
Conductor $75348$
Sign $-1$
Analytic cond. $601.656$
Root an. cond. $24.5286$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s + 3·11-s + 13-s − 19-s − 23-s + 4·25-s + 6·29-s − 4·31-s − 3·35-s − 10·37-s − 10·43-s + 6·47-s + 49-s + 12·53-s − 9·55-s + 6·59-s − 4·61-s − 3·65-s + 5·67-s + 2·73-s + 3·77-s + 14·79-s − 15·83-s + 6·89-s + 91-s + 3·95-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 0.904·11-s + 0.277·13-s − 0.229·19-s − 0.208·23-s + 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.507·35-s − 1.64·37-s − 1.52·43-s + 0.875·47-s + 1/7·49-s + 1.64·53-s − 1.21·55-s + 0.781·59-s − 0.512·61-s − 0.372·65-s + 0.610·67-s + 0.234·73-s + 0.341·77-s + 1.57·79-s − 1.64·83-s + 0.635·89-s + 0.104·91-s + 0.307·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75348\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \cdot 23\)
Sign: $-1$
Analytic conductor: \(601.656\)
Root analytic conductor: \(24.5286\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75348,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.35748081968942, −13.81639886961338, −13.44522539925128, −12.59472010423755, −12.15746610382605, −11.92405554999469, −11.32523514646118, −10.99167550454004, −10.29976961414159, −9.911629914989862, −9.001471004450647, −8.665713880847518, −8.287822004016159, −7.685106915002936, −7.065273417413511, −6.772229778314719, −6.078471781213578, −5.300482785654255, −4.839720930627466, −4.040868382805125, −3.830878206621972, −3.239839799233554, −2.364116221955884, −1.587078341121003, −0.8570444555565090, 0, 0.8570444555565090, 1.587078341121003, 2.364116221955884, 3.239839799233554, 3.830878206621972, 4.040868382805125, 4.839720930627466, 5.300482785654255, 6.078471781213578, 6.772229778314719, 7.065273417413511, 7.685106915002936, 8.287822004016159, 8.665713880847518, 9.001471004450647, 9.911629914989862, 10.29976961414159, 10.99167550454004, 11.32523514646118, 11.92405554999469, 12.15746610382605, 12.59472010423755, 13.44522539925128, 13.81639886961338, 14.35748081968942

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