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Properties

Label 2-1110-185.142-c1-0-12
Degree $2$
Conductor $1110$
Sign $0.476 - 0.879i$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−2.23 + 0.161i)5-s + (0.707 − 0.707i)6-s + (1.80 + 1.80i)7-s − i·8-s + 1.00i·9-s + (−0.161 − 2.23i)10-s − 1.06i·11-s + (0.707 + 0.707i)12-s − 5.30i·13-s + (−1.80 + 1.80i)14-s + (1.69 + 1.46i)15-s + 16-s − 0.822·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.997 + 0.0721i)5-s + (0.288 − 0.288i)6-s + (0.682 + 0.682i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.0510 − 0.705i)10-s − 0.320i·11-s + (0.204 + 0.204i)12-s − 1.47i·13-s + (−0.482 + 0.482i)14-s + (0.436 + 0.377i)15-s + 0.250·16-s − 0.199·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.476 - 0.879i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.476 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.079766241\)
\(L(\frac12)\) \(\approx\) \(1.079766241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (2.23 - 0.161i)T \)
37 \( 1 + (-2.17 - 5.68i)T \)
good7 \( 1 + (-1.80 - 1.80i)T + 7iT^{2} \)
11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 + 5.30iT - 13T^{2} \)
17 \( 1 + 0.822T + 17T^{2} \)
19 \( 1 + (-0.279 - 0.279i)T + 19iT^{2} \)
23 \( 1 - 6.71iT - 23T^{2} \)
29 \( 1 + (-1.25 + 1.25i)T - 29iT^{2} \)
31 \( 1 + (-3.59 - 3.59i)T + 31iT^{2} \)
41 \( 1 - 2.48iT - 41T^{2} \)
43 \( 1 + 9.11iT - 43T^{2} \)
47 \( 1 + (-6.22 - 6.22i)T + 47iT^{2} \)
53 \( 1 + (-1.25 + 1.25i)T - 53iT^{2} \)
59 \( 1 + (-10.0 - 10.0i)T + 59iT^{2} \)
61 \( 1 + (-5.97 - 5.97i)T + 61iT^{2} \)
67 \( 1 + (-2.68 + 2.68i)T - 67iT^{2} \)
71 \( 1 + 6.33T + 71T^{2} \)
73 \( 1 + (-9.12 - 9.12i)T + 73iT^{2} \)
79 \( 1 + (-10.8 - 10.8i)T + 79iT^{2} \)
83 \( 1 + (2.66 - 2.66i)T - 83iT^{2} \)
89 \( 1 + (-1.96 + 1.96i)T - 89iT^{2} \)
97 \( 1 + 2.76T + 97T^{2} \)
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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

−10.00523725196828026151713840154, −8.732635240958127254395170562617, −8.182886763653518828448829130280, −7.59146673389947599185399028794, −6.73755457263505889054179466370, −5.60907847395237607629279173492, −5.16848870008861700865970869897, −3.96322915144428438458251960732, −2.79473120294959572119818382125, −0.948766885399756653618827079866, 0.71240261151690206983417559000, 2.22384895273550349959044287925, 3.73103020442356105842270183697, 4.40396407932159213563581571096, 4.87493476049154227406366910884, 6.41104367443294126757366246400, 7.26263832710435752863659864653, 8.194414252998033096826795745624, 8.973984702676381723392841175475, 9.841350701391161933952481582859

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