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Properties

Label 2-1110-185.64-c1-0-31
Degree $2$
Conductor $1110$
Sign $-0.425 + 0.905i$
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 + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.581 + 2.15i)5-s + 0.999i·6-s + (1.29 − 0.746i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.57 − 1.58i)10-s − 4.99·11-s + (−0.866 − 0.499i)12-s + (−3.55 − 6.16i)13-s + 1.49i·14-s + (0.575 + 2.16i)15-s + (−0.5 + 0.866i)16-s + (−2.24 + 3.88i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.260 + 0.965i)5-s + 0.408i·6-s + (0.488 − 0.282i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.499 − 0.500i)10-s − 1.50·11-s + (−0.249 − 0.144i)12-s + (−0.987 − 1.70i)13-s + 0.399i·14-s + (0.148 + 0.557i)15-s + (−0.125 + 0.216i)16-s + (−0.544 + 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.425 + 0.905i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3587463063\)
\(L(\frac12)\) \(\approx\) \(0.3587463063\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.581 - 2.15i)T \)
37 \( 1 + (4.87 - 3.64i)T \)
good7 \( 1 + (-1.29 + 0.746i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + (3.55 + 6.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.24 - 3.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.68 + 1.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.83T + 23T^{2} \)
29 \( 1 + 8.24iT - 29T^{2} \)
31 \( 1 - 0.925iT - 31T^{2} \)
41 \( 1 + (-3.23 - 5.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.04T + 43T^{2} \)
47 \( 1 + 4.51iT - 47T^{2} \)
53 \( 1 + (3.66 + 2.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.55 + 2.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.38 + 4.83i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.5 - 6.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.259 - 0.448i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 + (0.836 - 0.482i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.83 + 4.52i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.79 - 3.92i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.48T + 97T^{2} \)
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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

−9.744403635298795684860251941437, −8.267239767340627651152422021359, −7.81260202400834470296406781929, −7.51940597273133369912555963046, −6.30733316834628033353668620370, −5.49936855692643363883499124281, −4.38965872547062640016956724393, −3.07727496423647926270857447083, −2.20523677576571744987040283070, −0.15348103410546913655934619543, 1.79122500153425072207919723121, 2.59562973196433886485164932797, 4.01853642521214960438892442744, 4.76539616702610738948033548948, 5.47156852783185176899727024974, 7.24443632438153330463492903037, 7.77982563269984195370891694307, 8.708505547561663826996052732321, 9.238507973021228687684856340860, 9.933445806614743947260645088822

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