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Properties

Label 2-1110-37.11-c1-0-13
Degree $2$
Conductor $1110$
Sign $0.775 + 0.630i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·6-s + (−2.34 + 4.07i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + 6.11·11-s + (−0.499 − 0.866i)12-s + (5.75 + 3.32i)13-s + 4.69i·14-s + (−0.866 + 0.499i)15-s + (−0.5 − 0.866i)16-s + (4.13 − 2.38i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s − 0.408i·6-s + (−0.888 + 1.53i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s + 1.84·11-s + (−0.144 − 0.249i)12-s + (1.59 + 0.921i)13-s + 1.25i·14-s + (−0.223 + 0.129i)15-s + (−0.125 − 0.216i)16-s + (1.00 − 0.579i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.775 + 0.630i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.775 + 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.530768254\)
\(L(\frac12)\) \(\approx\) \(2.530768254\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-3.40 - 5.04i)T \)
good7 \( 1 + (2.34 - 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 6.11T + 11T^{2} \)
13 \( 1 + (-5.75 - 3.32i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.13 + 2.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.76 + 2.74i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.04iT - 23T^{2} \)
29 \( 1 - 4.78iT - 29T^{2} \)
31 \( 1 + 0.756iT - 31T^{2} \)
41 \( 1 + (-2.34 + 4.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 5.21iT - 43T^{2} \)
47 \( 1 - 6.93T + 47T^{2} \)
53 \( 1 + (-1.42 - 2.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.36 - 3.67i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.28 + 1.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.03 + 1.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.71 - 4.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.93T + 73T^{2} \)
79 \( 1 + (-11.6 - 6.70i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.99 - 5.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (13.9 - 8.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.1iT - 97T^{2} \)
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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

−9.401276953649278965454680846649, −9.008681375517352505176421795882, −8.428006404523197077399502209177, −6.79269398949713545609890306777, −6.45046412421780156162705103235, −5.64611585914658138871345312080, −4.26083328226024204299027602910, −3.50031300905543564333751606596, −2.46103804454753303335077445525, −1.23105951677680693195525889138, 1.19646218923739448808505062752, 3.38677891999022967778345081747, 3.74625492255743665337190435963, 4.26959816689659217760445008440, 6.03916122159170775344224711177, 6.29815797293108002998408511835, 7.45479918407958523874823117330, 8.089698868782346004496318366689, 9.122901923548836754684737124697, 10.01444233050133979193263498390

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