Properties

Label 2-1110-5.4-c1-0-1
Degree $2$
Conductor $1110$
Sign $-0.948 - 0.316i$
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 i·3-s − 4-s + (0.707 − 2.12i)5-s + 6-s + 2.41i·7-s i·8-s − 9-s + (2.12 + 0.707i)10-s − 6.41·11-s + i·12-s + 5.24i·13-s − 2.41·14-s + (−2.12 − 0.707i)15-s + 16-s − 3.58i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.316 − 0.948i)5-s + 0.408·6-s + 0.912i·7-s − 0.353i·8-s − 0.333·9-s + (0.670 + 0.223i)10-s − 1.93·11-s + 0.288i·12-s + 1.45i·13-s − 0.645·14-s + (−0.547 − 0.182i)15-s + 0.250·16-s − 0.869i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.948 - 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4198620739\)
\(L(\frac12)\) \(\approx\) \(0.4198620739\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (-0.707 + 2.12i)T \)
37 \( 1 - iT \)
good7 \( 1 - 2.41iT - 7T^{2} \)
11 \( 1 + 6.41T + 11T^{2} \)
13 \( 1 - 5.24iT - 13T^{2} \)
17 \( 1 + 3.58iT - 17T^{2} \)
19 \( 1 - 2.17T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 3.65T + 31T^{2} \)
41 \( 1 + 6.58T + 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 - 11.8iT - 47T^{2} \)
53 \( 1 + 3.82iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 + 6.48T + 61T^{2} \)
67 \( 1 - 0.828iT - 67T^{2} \)
71 \( 1 + 9.41T + 71T^{2} \)
73 \( 1 - 2.17iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 6.89iT - 83T^{2} \)
89 \( 1 + 2.89T + 89T^{2} \)
97 \( 1 - 6.72iT - 97T^{2} \)
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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

−9.754114774114758465716544464276, −9.287277963639603930108243252247, −8.466558543613646462699700530370, −7.74203092218434566120257952732, −6.99255461976536827688969365086, −5.75028243834403966390769203499, −5.38773032673114205143703896661, −4.49686991598703664509884565010, −2.82872287498152258370242058337, −1.70346025938810581967365430448, 0.17281953492235326151670769701, 2.19457354981061130362456614644, 3.17461469679749977336425181653, 3.84722069413717081328236694406, 5.23976147055750973998115521099, 5.70072068444289118259050456784, 7.20304895068573087182937466285, 7.81842300123690586617645336177, 8.739993089050823972134375580122, 10.07578728508869633560703413116

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