Properties

Label 2-1110-37.36-c1-0-8
Degree $2$
Conductor $1110$
Sign $-0.675 - 0.737i$
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 − 1.30·7-s i·8-s + 9-s − 10-s + 0.690·11-s − 12-s − 2.69i·13-s − 1.30i·14-s + i·15-s + 16-s + 6.90i·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 − 0.495·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.208·11-s − 0.288·12-s − 0.746i·13-s − 0.350i·14-s + 0.258i·15-s + 0.250·16-s + 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.675 - 0.737i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.519892147\)
\(L(\frac12)\) \(\approx\) \(1.519892147\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.10 - 4.48i)T \)
good7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 - 0.690T + 11T^{2} \)
13 \( 1 + 2.69iT - 13T^{2} \)
17 \( 1 - 6.90iT - 17T^{2} \)
19 \( 1 - 2.69iT - 19T^{2} \)
23 \( 1 - 8.90iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 9.59iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8.21iT - 43T^{2} \)
47 \( 1 + 4.61T + 47T^{2} \)
53 \( 1 - 0.690T + 53T^{2} \)
59 \( 1 + 6.21iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 - 5.92iT - 89T^{2} \)
97 \( 1 - 3.38iT - 97T^{2} \)
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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

−10.04588279933314269737514205377, −9.248366886282480204063696603828, −8.287788383643369800637694671990, −7.78808809279929974318178123569, −6.79340467680432250309696574288, −6.08961524103645144034643711267, −5.16724188425490238388222900796, −3.77142648111712192071763491712, −3.27379169299567488510848209138, −1.63653399655713109092775493772, 0.63660432305458623611981268130, 2.21650449288896423544958415927, 3.01963307213901217309815872341, 4.25342075985463205402851253593, 4.83304020450057745203569416331, 6.20363989591515745785422576131, 7.11846821706763686852109765434, 8.093928667592776602905139561298, 9.077454236990862449874736293072, 9.385312127071128340443376349399

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