Properties

Label 2-1110-185.142-c1-0-8
Degree $2$
Conductor $1110$
Sign $-0.897 - 0.441i$
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 + (1.88 + 1.19i)5-s + (−0.707 + 0.707i)6-s + (−1.50 − 1.50i)7-s i·8-s + 1.00i·9-s + (−1.19 + 1.88i)10-s + 0.249i·11-s + (−0.707 − 0.707i)12-s + 2.83i·13-s + (1.50 − 1.50i)14-s + (0.486 + 2.18i)15-s + 16-s − 7.29·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (0.843 + 0.536i)5-s + (−0.288 + 0.288i)6-s + (−0.569 − 0.569i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.379 + 0.596i)10-s + 0.0750i·11-s + (−0.204 − 0.204i)12-s + 0.785i·13-s + (0.402 − 0.402i)14-s + (0.125 + 0.563i)15-s + 0.250·16-s − 1.76·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.897 - 0.441i$
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: \((2,\ 1110,\ (\ :1/2),\ -0.897 - 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.550397382\)
\(L(\frac12)\) \(\approx\) \(1.550397382\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 1.19i)T \)
37 \( 1 + (-3.06 + 5.25i)T \)
good7 \( 1 + (1.50 + 1.50i)T + 7iT^{2} \)
11 \( 1 - 0.249iT - 11T^{2} \)
13 \( 1 - 2.83iT - 13T^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 + (-1.89 - 1.89i)T + 19iT^{2} \)
23 \( 1 - 8.28iT - 23T^{2} \)
29 \( 1 + (2.37 - 2.37i)T - 29iT^{2} \)
31 \( 1 + (-1.83 - 1.83i)T + 31iT^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 7.61iT - 43T^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (-4.20 + 4.20i)T - 53iT^{2} \)
59 \( 1 + (-6.26 - 6.26i)T + 59iT^{2} \)
61 \( 1 + (3.22 + 3.22i)T + 61iT^{2} \)
67 \( 1 + (-5.68 + 5.68i)T - 67iT^{2} \)
71 \( 1 - 3.37T + 71T^{2} \)
73 \( 1 + (7.48 + 7.48i)T + 73iT^{2} \)
79 \( 1 + (-5.99 - 5.99i)T + 79iT^{2} \)
83 \( 1 + (-5.98 + 5.98i)T - 83iT^{2} \)
89 \( 1 + (-5.37 + 5.37i)T - 89iT^{2} \)
97 \( 1 + 5.94T + 97T^{2} \)
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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

−9.747419987605854055803152998025, −9.549350640011765648483255343176, −8.668448309773174873290898755488, −7.53279937158269740402100537461, −6.82871718619604034481863009576, −6.16358219010872699289292996862, −5.11200410598659644954597762957, −4.10434135249204026678969533656, −3.16263771028969537300470129871, −1.81577798603589130579050024717, 0.62857953605653630307369516030, 2.23003313185781589363529135746, 2.68969939861315460337334592354, 4.11159907246212023731196184794, 5.13649681451480286314736042718, 6.08976899713966693314969806823, 6.85656074171623141410605138276, 8.226004090104069286540937253946, 8.840782687954443874993244259962, 9.393895895503877232864329607770

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