Properties

Label 2-155848-1.1-c1-0-2
Degree $2$
Conductor $155848$
Sign $1$
Analytic cond. $1244.45$
Root an. cond. $35.2767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 7-s + 9-s + 6·17-s − 6·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 6·29-s + 8·31-s + 2·37-s + 2·41-s + 8·43-s − 8·47-s + 49-s − 12·51-s − 2·53-s + 12·57-s − 6·59-s − 63-s − 12·67-s − 2·69-s − 8·71-s + 6·73-s + 10·75-s + 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.45·17-s − 1.37·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 1.68·51-s − 0.274·53-s + 1.58·57-s − 0.781·59-s − 0.125·63-s − 1.46·67-s − 0.240·69-s − 0.949·71-s + 0.702·73-s + 1.15·75-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(155848\)    =    \(2^{3} \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1244.45\)
Root analytic conductor: \(35.2767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 155848,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088472215\)
\(L(\frac12)\) \(\approx\) \(1.088472215\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.24549133871715, −12.62765319243998, −12.23135181749915, −11.99480836062590, −11.45494786621351, −10.91204134592195, −10.39825167829313, −10.20729808405112, −9.539592736791400, −9.098816000083687, −8.312662076138275, −8.014357971090868, −7.467181575133010, −6.679445776619944, −6.325521994376455, −6.019756464443974, −5.450383230708389, −4.885205312119928, −4.370489399341359, −3.864669135992683, −2.972550353289681, −2.691685071782665, −1.703007528722514, −1.017494718173147, −0.3882928899538985, 0.3882928899538985, 1.017494718173147, 1.703007528722514, 2.691685071782665, 2.972550353289681, 3.864669135992683, 4.370489399341359, 4.885205312119928, 5.450383230708389, 6.019756464443974, 6.325521994376455, 6.679445776619944, 7.467181575133010, 8.014357971090868, 8.312662076138275, 9.098816000083687, 9.539592736791400, 10.20729808405112, 10.39825167829313, 10.91204134592195, 11.45494786621351, 11.99480836062590, 12.23135181749915, 12.62765319243998, 13.24549133871715

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