Properties

Label 2-88935-1.1-c1-0-35
Degree $2$
Conductor $88935$
Sign $-1$
Analytic cond. $710.149$
Root an. cond. $26.6486$
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  + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 9-s − 2·10-s − 2·12-s − 6·13-s + 15-s − 4·16-s − 7·17-s + 2·18-s − 5·19-s − 2·20-s − 23-s + 25-s − 12·26-s − 27-s + 5·29-s + 2·30-s + 8·31-s − 8·32-s − 14·34-s + 2·36-s − 2·37-s − 10·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 0.577·12-s − 1.66·13-s + 0.258·15-s − 16-s − 1.69·17-s + 0.471·18-s − 1.14·19-s − 0.447·20-s − 0.208·23-s + 1/5·25-s − 2.35·26-s − 0.192·27-s + 0.928·29-s + 0.365·30-s + 1.43·31-s − 1.41·32-s − 2.40·34-s + 1/3·36-s − 0.328·37-s − 1.62·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(88935\)    =    \(3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(710.149\)
Root analytic conductor: \(26.6486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{88935} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 88935,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−14.18403225880023, −13.67263337450993, −12.82712226340988, −12.76946035305286, −12.34116203021004, −11.81010835814535, −11.26389567694204, −10.97480568661053, −10.36318635521637, −9.649999137243738, −9.267757562909760, −8.482212533027949, −8.037777249724161, −7.228603904415850, −6.831575601130531, −6.304089544344336, −5.985873856689669, −5.002047561574394, −4.831638625681966, −4.284227429694364, −3.995567366567366, −2.989172137834768, −2.435738453570372, −2.096082363096853, −0.7228156947500008, 0, 0.7228156947500008, 2.096082363096853, 2.435738453570372, 2.989172137834768, 3.995567366567366, 4.284227429694364, 4.831638625681966, 5.002047561574394, 5.985873856689669, 6.304089544344336, 6.831575601130531, 7.228603904415850, 8.037777249724161, 8.482212533027949, 9.267757562909760, 9.649999137243738, 10.36318635521637, 10.97480568661053, 11.26389567694204, 11.81010835814535, 12.34116203021004, 12.76946035305286, 12.82712226340988, 13.67263337450993, 14.18403225880023

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