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Properties

Label 2-326700-1.1-c1-0-129
Degree $2$
Conductor $326700$
Sign $-1$
Analytic cond. $2608.71$
Root an. cond. $51.0755$
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  − 4·7-s + 5·13-s + 19-s + 11·31-s + 37-s + 8·43-s + 9·49-s + 13·61-s + 16·67-s − 10·73-s + 4·79-s − 20·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.51·7-s + 1.38·13-s + 0.229·19-s + 1.97·31-s + 0.164·37-s + 1.21·43-s + 9/7·49-s + 1.66·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s − 2.09·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(326700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(2608.71\)
Root analytic conductor: \(51.0755\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 326700,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.86250162851920, −12.47586797319510, −11.98333139634806, −11.39257427347417, −11.11810382485074, −10.43279299137437, −10.10870869260594, −9.662399453633723, −9.244544754458528, −8.728360265439167, −8.266236263648501, −7.877076851123835, −7.114958539823117, −6.658043207089988, −6.429191573785297, −5.798770618997876, −5.548811141825750, −4.715004435006800, −4.140847300579289, −3.701347468387505, −3.244640774753684, −2.672966075532914, −2.227173113762841, −1.149322284211527, −0.8807505699413323, 0, 0.8807505699413323, 1.149322284211527, 2.227173113762841, 2.672966075532914, 3.244640774753684, 3.701347468387505, 4.140847300579289, 4.715004435006800, 5.548811141825750, 5.798770618997876, 6.429191573785297, 6.658043207089988, 7.114958539823117, 7.877076851123835, 8.266236263648501, 8.728360265439167, 9.244544754458528, 9.662399453633723, 10.10870869260594, 10.43279299137437, 11.11810382485074, 11.39257427347417, 11.98333139634806, 12.47586797319510, 12.86250162851920

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