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Properties

Label 2-1110-185.117-c1-0-25
Degree $2$
Conductor $1110$
Sign $0.584 - 0.811i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.707 + 0.707i)3-s + 4-s + (1.43 + 1.71i)5-s + (0.707 + 0.707i)6-s + (3.31 + 3.31i)7-s + 8-s + 1.00i·9-s + (1.43 + 1.71i)10-s − 4.55i·11-s + (0.707 + 0.707i)12-s − 0.0409·13-s + (3.31 + 3.31i)14-s + (−0.199 + 2.22i)15-s + 16-s − 3.39i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (0.641 + 0.767i)5-s + (0.288 + 0.288i)6-s + (1.25 + 1.25i)7-s + 0.353·8-s + 0.333i·9-s + (0.453 + 0.542i)10-s − 1.37i·11-s + (0.204 + 0.204i)12-s − 0.0113·13-s + (0.885 + 0.885i)14-s + (−0.0514 + 0.575i)15-s + 0.250·16-s − 0.822i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.571345343\)
\(L(\frac12)\) \(\approx\) \(3.571345343\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.43 - 1.71i)T \)
37 \( 1 + (5.93 + 1.32i)T \)
good7 \( 1 + (-3.31 - 3.31i)T + 7iT^{2} \)
11 \( 1 + 4.55iT - 11T^{2} \)
13 \( 1 + 0.0409T + 13T^{2} \)
17 \( 1 + 3.39iT - 17T^{2} \)
19 \( 1 + (-1.42 + 1.42i)T - 19iT^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + (2.73 + 2.73i)T + 29iT^{2} \)
31 \( 1 + (-2.17 + 2.17i)T - 31iT^{2} \)
41 \( 1 + 4.17iT - 41T^{2} \)
43 \( 1 + 8.29T + 43T^{2} \)
47 \( 1 + (-1.79 - 1.79i)T + 47iT^{2} \)
53 \( 1 + (5.35 - 5.35i)T - 53iT^{2} \)
59 \( 1 + (-7.59 + 7.59i)T - 59iT^{2} \)
61 \( 1 + (-4.71 + 4.71i)T - 61iT^{2} \)
67 \( 1 + (10.3 - 10.3i)T - 67iT^{2} \)
71 \( 1 - 3.26T + 71T^{2} \)
73 \( 1 + (3.99 + 3.99i)T + 73iT^{2} \)
79 \( 1 + (-3.34 + 3.34i)T - 79iT^{2} \)
83 \( 1 + (-1.21 + 1.21i)T - 83iT^{2} \)
89 \( 1 + (-9.81 - 9.81i)T + 89iT^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−10.02231062203534868748096263374, −9.104607009311060483387345866220, −8.341519710022654700859325534849, −7.54723895278682933001023954661, −6.28748768890288908757911959554, −5.60094498443193732713462507232, −4.96994464125971416521360241084, −3.65532158002730872792830159814, −2.70430946296709802403515260342, −1.93772145984181806599828711899, 1.48357252695358923420249920536, 1.95637372563471572540350653097, 3.73068164896906140706388221572, 4.53048572369302942594701596452, 5.19571196279741896991268688337, 6.35447871746338157679305085577, 7.28623672956496846732357042998, 7.932633730740710573769893854486, 8.692233035290368520589261542230, 10.06826181468824137926930420612

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