Properties

Label 2-1110-1.1-c5-0-103
Degree $2$
Conductor $1110$
Sign $-1$
Analytic cond. $178.026$
Root an. cond. $13.3426$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s + 25·5-s − 36·6-s + 33·7-s + 64·8-s + 81·9-s + 100·10-s + 39·11-s − 144·12-s − 314·13-s + 132·14-s − 225·15-s + 256·16-s − 1.94e3·17-s + 324·18-s + 922·19-s + 400·20-s − 297·21-s + 156·22-s + 3.84e3·23-s − 576·24-s + 625·25-s − 1.25e3·26-s − 729·27-s + 528·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.254·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.0971·11-s − 0.288·12-s − 0.515·13-s + 0.179·14-s − 0.258·15-s + 1/4·16-s − 1.63·17-s + 0.235·18-s + 0.585·19-s + 0.223·20-s − 0.146·21-s + 0.0687·22-s + 1.51·23-s − 0.204·24-s + 1/5·25-s − 0.364·26-s − 0.192·27-s + 0.127·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(178.026\)
Root analytic conductor: \(13.3426\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1110,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 + p^{2} T \)
5 \( 1 - p^{2} T \)
37 \( 1 + p^{2} T \)
good7 \( 1 - 33 T + p^{5} T^{2} \)
11 \( 1 - 39 T + p^{5} T^{2} \)
13 \( 1 + 314 T + p^{5} T^{2} \)
17 \( 1 + 1949 T + p^{5} T^{2} \)
19 \( 1 - 922 T + p^{5} T^{2} \)
23 \( 1 - 3844 T + p^{5} T^{2} \)
29 \( 1 + 8313 T + p^{5} T^{2} \)
31 \( 1 - 5985 T + p^{5} T^{2} \)
41 \( 1 + 13607 T + p^{5} T^{2} \)
43 \( 1 - 847 T + p^{5} T^{2} \)
47 \( 1 - 2904 T + p^{5} T^{2} \)
53 \( 1 + 33851 T + p^{5} T^{2} \)
59 \( 1 + 2186 T + p^{5} T^{2} \)
61 \( 1 - 19893 T + p^{5} T^{2} \)
67 \( 1 + 18596 T + p^{5} T^{2} \)
71 \( 1 + 17740 T + p^{5} T^{2} \)
73 \( 1 + 44536 T + p^{5} T^{2} \)
79 \( 1 - 79732 T + p^{5} T^{2} \)
83 \( 1 - 36254 T + p^{5} T^{2} \)
89 \( 1 + 57970 T + p^{5} T^{2} \)
97 \( 1 - 85531 T + p^{5} T^{2} \)
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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

−8.817625449677982462190695239139, −7.58972333863139846140215361084, −6.82287394271354722154467743141, −6.14266795133016546948277594002, −5.07916334086974064967327647066, −4.69365901533234397167874136417, −3.44822094613482913873296857996, −2.33648225453125840007258238095, −1.37313653918855264984020667891, 0, 1.37313653918855264984020667891, 2.33648225453125840007258238095, 3.44822094613482913873296857996, 4.69365901533234397167874136417, 5.07916334086974064967327647066, 6.14266795133016546948277594002, 6.82287394271354722154467743141, 7.58972333863139846140215361084, 8.817625449677982462190695239139

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