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Properties

Label 2-1110-185.142-c1-0-3
Degree $2$
Conductor $1110$
Sign $-0.0180 - 0.999i$
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 + (−0.707 − 0.707i)3-s − 4-s + (−2.07 − 0.837i)5-s + (0.707 − 0.707i)6-s + (−2.93 − 2.93i)7-s − i·8-s + 1.00i·9-s + (0.837 − 2.07i)10-s + 1.12i·11-s + (0.707 + 0.707i)12-s + 2.10i·13-s + (2.93 − 2.93i)14-s + (0.874 + 2.05i)15-s + 16-s − 3.81·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.927 − 0.374i)5-s + (0.288 − 0.288i)6-s + (−1.11 − 1.11i)7-s − 0.353i·8-s + 0.333i·9-s + (0.264 − 0.655i)10-s + 0.340i·11-s + (0.204 + 0.204i)12-s + 0.583i·13-s + (0.785 − 0.785i)14-s + (0.225 + 0.531i)15-s + 0.250·16-s − 0.926·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0180 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0180 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.0180 - 0.999i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.0180 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5013745646\)
\(L(\frac12)\) \(\approx\) \(0.5013745646\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 + 0.837i)T \)
37 \( 1 + (-5.91 - 1.41i)T \)
good7 \( 1 + (2.93 + 2.93i)T + 7iT^{2} \)
11 \( 1 - 1.12iT - 11T^{2} \)
13 \( 1 - 2.10iT - 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + (0.0424 + 0.0424i)T + 19iT^{2} \)
23 \( 1 + 7.44iT - 23T^{2} \)
29 \( 1 + (-1.00 + 1.00i)T - 29iT^{2} \)
31 \( 1 + (-5.55 - 5.55i)T + 31iT^{2} \)
41 \( 1 - 7.31iT - 41T^{2} \)
43 \( 1 - 6.59iT - 43T^{2} \)
47 \( 1 + (-3.52 - 3.52i)T + 47iT^{2} \)
53 \( 1 + (6.43 - 6.43i)T - 53iT^{2} \)
59 \( 1 + (-1.53 - 1.53i)T + 59iT^{2} \)
61 \( 1 + (-3.13 - 3.13i)T + 61iT^{2} \)
67 \( 1 + (-2.59 + 2.59i)T - 67iT^{2} \)
71 \( 1 - 2.10T + 71T^{2} \)
73 \( 1 + (6.95 + 6.95i)T + 73iT^{2} \)
79 \( 1 + (8.44 + 8.44i)T + 79iT^{2} \)
83 \( 1 + (2.80 - 2.80i)T - 83iT^{2} \)
89 \( 1 + (8.09 - 8.09i)T - 89iT^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−10.01238968064144032350441053791, −9.084312880032676944946006094132, −8.217292764198563464910233167195, −7.40877071397896186739530922675, −6.67472557268151412319326831854, −6.26359038267906358870180377897, −4.52669784097054187260737332639, −4.41514800217115166758462968637, −2.98592304974249588401316652127, −0.910243413548004922742034997487, 0.31697348950504972946145365622, 2.47291962920828745434265081908, 3.33445802648622879179822601170, 4.10475548384286496763863680834, 5.33726785387807503550413443762, 6.08831342558755351456032688290, 7.08759672033117912852374628072, 8.230520113856321875825042158543, 8.991477042444933309997086770438, 9.743922257274942166359610803796

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