Properties

Label 2-4410-1.1-c1-0-45
Degree $2$
Conductor $4410$
Sign $-1$
Analytic cond. $35.2140$
Root an. cond. $5.93414$
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-s + 4-s − 5-s − 8-s + 10-s − 4·11-s + 6·13-s + 16-s + 2·17-s − 20-s + 4·22-s + 25-s − 6·26-s − 6·29-s − 8·31-s − 32-s − 2·34-s − 10·37-s + 40-s + 2·41-s + 4·43-s − 4·44-s + 8·47-s − 50-s + 6·52-s + 2·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 1.20·11-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1.64·37-s + 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 1.16·47-s − 0.141·50-s + 0.832·52-s + 0.274·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(35.2140\)
Root analytic conductor: \(5.93414\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4410} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.957237096013159459080835086230, −7.50793050530402086317500860857, −6.71654507983987973017676739321, −5.73468162934481500102414565629, −5.29992032847947121483959754241, −3.94449175269007332055216124286, −3.39915186610779589390347542849, −2.30609205239508517525684039531, −1.25635141165548889358757283011, 0, 1.25635141165548889358757283011, 2.30609205239508517525684039531, 3.39915186610779589390347542849, 3.94449175269007332055216124286, 5.29992032847947121483959754241, 5.73468162934481500102414565629, 6.71654507983987973017676739321, 7.50793050530402086317500860857, 7.957237096013159459080835086230

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