Properties

Label 2-1110-1.1-c3-0-19
Degree $2$
Conductor $1110$
Sign $1$
Analytic cond. $65.4921$
Root an. cond. $8.09272$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 5·5-s − 6·6-s + 10·7-s + 8·8-s + 9·9-s − 10·10-s + 44·11-s − 12·12-s + 59·13-s + 20·14-s + 15·15-s + 16·16-s − 46·17-s + 18·18-s − 34·19-s − 20·20-s − 30·21-s + 88·22-s + 6·23-s − 24·24-s + 25·25-s + 118·26-s − 27·27-s + 40·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.539·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 1.25·13-s + 0.381·14-s + 0.258·15-s + 1/4·16-s − 0.656·17-s + 0.235·18-s − 0.410·19-s − 0.223·20-s − 0.311·21-s + 0.852·22-s + 0.0543·23-s − 0.204·24-s + 1/5·25-s + 0.890·26-s − 0.192·27-s + 0.269·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(65.4921\)
Root analytic conductor: \(8.09272\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.104826695\)
\(L(\frac12)\) \(\approx\) \(3.104826695\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 + p T \)
5 \( 1 + p T \)
37 \( 1 - p T \)
good7 \( 1 - 10 T + p^{3} T^{2} \)
11 \( 1 - 4 p T + p^{3} T^{2} \)
13 \( 1 - 59 T + p^{3} T^{2} \)
17 \( 1 + 46 T + p^{3} T^{2} \)
19 \( 1 + 34 T + p^{3} T^{2} \)
23 \( 1 - 6 T + p^{3} T^{2} \)
29 \( 1 - 7 T + p^{3} T^{2} \)
31 \( 1 + 182 T + p^{3} T^{2} \)
41 \( 1 - 360 T + p^{3} T^{2} \)
43 \( 1 - 101 T + p^{3} T^{2} \)
47 \( 1 + 35 T + p^{3} T^{2} \)
53 \( 1 + 507 T + p^{3} T^{2} \)
59 \( 1 - 821 T + p^{3} T^{2} \)
61 \( 1 - 70 T + p^{3} T^{2} \)
67 \( 1 - 612 T + p^{3} T^{2} \)
71 \( 1 - 88 T + p^{3} T^{2} \)
73 \( 1 - 622 T + p^{3} T^{2} \)
79 \( 1 - 8 T + p^{3} T^{2} \)
83 \( 1 - 1223 T + p^{3} T^{2} \)
89 \( 1 - 345 T + p^{3} T^{2} \)
97 \( 1 - 870 T + p^{3} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.423024744116603986507452653974, −8.604932737340222053356867888516, −7.67744513347574988284630534461, −6.65620952767566269563834297947, −6.15374719408614556561358194587, −5.09652429148215084293647153013, −4.17569626603987936486492610908, −3.59545060263045443917002701630, −1.98852534894635374816751593765, −0.894283755977934126501849682698, 0.894283755977934126501849682698, 1.98852534894635374816751593765, 3.59545060263045443917002701630, 4.17569626603987936486492610908, 5.09652429148215084293647153013, 6.15374719408614556561358194587, 6.65620952767566269563834297947, 7.67744513347574988284630534461, 8.604932737340222053356867888516, 9.423024744116603986507452653974

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