Properties

Label 2-1110-185.184-c1-0-30
Degree $2$
Conductor $1110$
Sign $-0.294 + 0.955i$
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  + 2-s + i·3-s + 4-s + (−2 − i)5-s + i·6-s − 3i·7-s + 8-s − 9-s + (−2 − i)10-s − 3·11-s + i·12-s + 4·13-s − 3i·14-s + (1 − 2i)15-s + 16-s − 7·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408i·6-s − 1.13i·7-s + 0.353·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.904·11-s + 0.288i·12-s + 1.10·13-s − 0.801i·14-s + (0.258 − 0.516i)15-s + 0.250·16-s − 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.294 + 0.955i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.294 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270573610\)
\(L(\frac12)\) \(\approx\) \(1.270573610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 + (2 + i)T \)
37 \( 1 + (6 - i)T \)
good7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 7T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 5iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.734147090374738336236109399737, −8.610783437927459451696537297198, −7.959606861417185467106879037706, −7.06911250900582908531332821944, −6.12964104100658846555451893656, −5.00294905636482153606728987648, −4.13407548213147831280632398367, −3.84664529118434080072232867263, −2.38792154522695698696898176337, −0.41617444926428710550282091252, 1.89667312577178878410517798237, 2.89766463121113304753673570649, 3.83205948541746557387053060939, 4.97212452293138690184969758999, 5.95474126877153250044088212120, 6.58280040404106115335816011621, 7.55055457524199038899952707646, 8.366668653066126170496621987381, 8.928070521712944586832036579470, 10.50481981018230467959420619220

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