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Properties

Label 2-1110-37.11-c1-0-15
Degree $2$
Conductor $1110$
Sign $0.947 + 0.319i$
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 + (1.60 − 2.77i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + 2.56·11-s + (−0.499 − 0.866i)12-s + (4.60 + 2.65i)13-s + 3.20i·14-s + (0.866 − 0.499i)15-s + (−0.5 − 0.866i)16-s + (−5.12 + 2.96i)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 + (0.604 − 1.04i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s + 0.772·11-s + (−0.144 − 0.249i)12-s + (1.27 + 0.736i)13-s + 0.855i·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (−1.24 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.947 + 0.319i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.947 + 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.677290491\)
\(L(\frac12)\) \(\approx\) \(1.677290491\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.95 - 3.53i)T \)
good7 \( 1 + (-1.60 + 2.77i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.56T + 11T^{2} \)
13 \( 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.12 - 2.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.09 - 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.37iT - 23T^{2} \)
29 \( 1 + 3.05iT - 29T^{2} \)
31 \( 1 + 0.762iT - 31T^{2} \)
41 \( 1 + (0.168 - 0.292i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.72iT - 43T^{2} \)
47 \( 1 + 8.59T + 47T^{2} \)
53 \( 1 + (-5.08 - 8.81i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.08 + 3.51i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.3 + 6.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.31 + 7.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.79 - 3.11i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + (-13.5 - 7.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0857 + 0.148i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.85 + 3.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−9.660648682559473523934588408219, −8.852842712165932880122033884956, −8.183655931389469080080048846611, −7.32653759038118021580044471615, −6.55374685958793147920611600162, −6.00051602912183732815056770607, −4.50124641055847177598032016370, −3.60609371752333274348187070964, −1.94646969292467427671145574544, −1.12804330585259286455127055609, 1.25805844368842946084617007395, 2.47725306969849162596152694633, 3.43701853118079344992655946341, 4.69085821486809525769522956215, 5.56452689068977055563297448843, 6.53420217450046008132507760208, 7.69840902468527202235545188733, 8.631344860223265342384859442039, 9.047082688681633304291707627592, 9.625734807855725364674501815358

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