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Properties

Label 2-1110-37.36-c1-0-18
Degree $2$
Conductor $1110$
Sign $0.954 + 0.299i$
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 − 4.26·7-s − i·8-s + 9-s + 10-s + 5.24·11-s − 12-s − 1.82i·13-s − 4.26i·14-s − i·15-s + 16-s + 2.26i·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 − 1.61·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.58·11-s − 0.288·12-s − 0.505i·13-s − 1.14i·14-s − 0.258i·15-s + 0.250·16-s + 0.550i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.954 + 0.299i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552886637\)
\(L(\frac12)\) \(\approx\) \(1.552886637\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.80 - 1.82i)T \)
good7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 - 5.24T + 11T^{2} \)
13 \( 1 + 1.82iT - 13T^{2} \)
17 \( 1 - 2.26iT - 17T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 + 1.82iT - 23T^{2} \)
29 \( 1 + 10.0iT - 29T^{2} \)
31 \( 1 + 4.09iT - 31T^{2} \)
41 \( 1 - 4.09T + 41T^{2} \)
43 \( 1 - 0.446iT - 43T^{2} \)
47 \( 1 - 9.87T + 47T^{2} \)
53 \( 1 + 2.26T + 53T^{2} \)
59 \( 1 + 3.64iT - 59T^{2} \)
61 \( 1 - 2.53iT - 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + 4.71T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 4.17iT - 89T^{2} \)
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

−9.436347122669875381126151368431, −9.131250271393066855891813732448, −8.198223273054487748816977890933, −7.24775487850754924354675438159, −6.38505115699869268724278276823, −5.92676007572138601194018874297, −4.38809682778825628547190291771, −3.78038841065032231900085266698, −2.60696791974846211043663761359, −0.71282682693049078513423651912, 1.35741105328725901855168827261, 2.74263697392464393422584078254, 3.54711612946183544983508582897, 4.13618570383859502722725238991, 5.72076850171213186272904973566, 6.67699076169673721236026550675, 7.22475169265571844147736901086, 8.600559938690159693753572087586, 9.346931864764021605572588897749, 9.676029559502405650077614896901

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