Properties

Label 2-1110-5.4-c1-0-27
Degree $2$
Conductor $1110$
Sign $-0.964 - 0.265i$
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 i·3-s − 4-s + (−0.594 + 2.15i)5-s − 6-s + 1.17i·7-s + i·8-s − 9-s + (2.15 + 0.594i)10-s − 0.822·11-s + i·12-s − 1.74i·13-s + 1.17·14-s + (2.15 + 0.594i)15-s + 16-s − 3.94i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.265 + 0.964i)5-s − 0.408·6-s + 0.445i·7-s + 0.353i·8-s − 0.333·9-s + (0.681 + 0.187i)10-s − 0.247·11-s + 0.288i·12-s − 0.485i·13-s + 0.314·14-s + (0.556 + 0.153i)15-s + 0.250·16-s − 0.956i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4229635403\)
\(L(\frac12)\) \(\approx\) \(0.4229635403\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (0.594 - 2.15i)T \)
37 \( 1 - iT \)
good7 \( 1 - 1.17iT - 7T^{2} \)
11 \( 1 + 0.822T + 11T^{2} \)
13 \( 1 + 1.74iT - 13T^{2} \)
17 \( 1 + 3.94iT - 17T^{2} \)
19 \( 1 + 1.29T + 19T^{2} \)
23 \( 1 + 7.68iT - 23T^{2} \)
29 \( 1 + 9.23T + 29T^{2} \)
31 \( 1 + 9.30T + 31T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 2.92iT - 43T^{2} \)
47 \( 1 - 7.18iT - 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 + 3.18T + 59T^{2} \)
61 \( 1 - 8.78T + 61T^{2} \)
67 \( 1 + 15.2iT - 67T^{2} \)
71 \( 1 + 6.65T + 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 - 0.797T + 79T^{2} \)
83 \( 1 + 7.72iT - 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 13.0iT - 97T^{2} \)
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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

−9.490711548029505911507434267565, −8.608997374787592777465131984633, −7.72386575922939745142261876458, −6.99595446092550162457999984939, −6.01221144591527665361670341924, −5.08201040293757682389345104764, −3.77567590844139524435815484987, −2.82313477782778467537032769892, −2.04603127975099496166431020128, −0.18010721482245116791057655205, 1.66217046851468945730540157287, 3.72586691924351620731351173842, 4.12532904581334507702006683346, 5.34314020679153977273731953578, 5.76147505747539516801835042089, 7.17580412668296364084963805995, 7.74768781357659008225257939690, 8.791374236878671233133090701168, 9.199665417964938960289680415430, 10.10672850703432983000067091951

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