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Properties

Label 2-1110-37.26-c1-0-5
Degree $2$
Conductor $1110$
Sign $-0.831 - 0.555i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s + (−1 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 5·11-s + (0.499 + 0.866i)12-s + (−2.5 + 4.33i)13-s − 1.99·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s + (−0.377 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.50·11-s + (0.144 + 0.249i)12-s + (−0.693 + 1.20i)13-s − 0.534·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.831 - 0.555i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.831 - 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.064039995\)
\(L(\frac12)\) \(\approx\) \(1.064039995\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 6.06i)T \)
good7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 9T + 47T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−9.895229605154386705753371611054, −9.258885623341698062529608626560, −8.279524988625482730407823827717, −7.79388202743721958840015857397, −6.79331068334586798801399775171, −5.92412104781530460098390377473, −5.25889641366113415656529714713, −4.17533733727464206227780713276, −2.91216122600353537452807139657, −1.90660766179344916812031405289, 0.36968190955454456474643329672, 2.46533774132156703503218947021, 3.01787900420208888078971942426, 4.07700134980005773011811315758, 5.22187997594200934365590020570, 5.67999674593804376707799382122, 7.27202117645734074763766748881, 7.67797784157869321364174352747, 8.952677386355150180209916458371, 9.830617483174062762974913241447

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