Properties

Label 2-1110-37.36-c1-0-23
Degree $2$
Conductor $1110$
Sign $0.0326 + 0.999i$
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 + 3-s − 4-s + i·5-s + i·6-s − 4.44·7-s i·8-s + 9-s − 10-s − 2.44·11-s − 12-s + 0.440i·13-s − 4.44i·14-s + i·15-s + 16-s − 4.83i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s − 1.67·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.735·11-s − 0.288·12-s + 0.122i·13-s − 1.18i·14-s + 0.258i·15-s + 0.250·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.0326 + 0.999i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.0326 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4269868788\)
\(L(\frac12)\) \(\approx\) \(0.4269868788\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - iT \)
37 \( 1 + (0.198 + 6.07i)T \)
good7 \( 1 + 4.44T + 7T^{2} \)
11 \( 1 + 2.44T + 11T^{2} \)
13 \( 1 - 0.440iT - 13T^{2} \)
17 \( 1 + 4.83iT - 17T^{2} \)
19 \( 1 + 0.440iT - 19T^{2} \)
23 \( 1 + 2.83iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 5.27iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 0.396iT - 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 2.44T + 53T^{2} \)
59 \( 1 - 2.39iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 - 5.71T + 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 + 7.71T + 83T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 + 2.88iT - 97T^{2} \)
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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

−9.642282284411137734600918924071, −8.858243249905826382408760401069, −7.84238938672044003310368886005, −7.10068981641414724842849812553, −6.45914230597440256840118688956, −5.57015719843406120080524538559, −4.35748155591622308533367233526, −3.27456400561402474595884393913, −2.55380178145259060831776332519, −0.16752176211543677570180018239, 1.60928711298272813010075349699, 3.00128140812606589667031366775, 3.48534724967726944665300173346, 4.65028642070425191317479203836, 5.77313572608271397040835871856, 6.67558134653437033217405456223, 7.78450579861553867267090221853, 8.628086987810301166379856845241, 9.295820885113401048653747054865, 10.14534018266567159340487372193

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