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Properties

Label 2-1110-185.43-c1-0-32
Degree $2$
Conductor $1110$
Sign $0.401 + 0.915i$
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 + (−0.0484 − 2.23i)5-s + (0.707 + 0.707i)6-s + (2.77 − 2.77i)7-s − i·8-s − 1.00i·9-s + (2.23 − 0.0484i)10-s + 2.74i·11-s + (−0.707 + 0.707i)12-s − 2.09i·13-s + (2.77 + 2.77i)14-s + (−1.61 − 1.54i)15-s + 16-s − 4.54·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.0216 − 0.999i)5-s + (0.288 + 0.288i)6-s + (1.04 − 1.04i)7-s − 0.353i·8-s − 0.333i·9-s + (0.706 − 0.0153i)10-s + 0.826i·11-s + (−0.204 + 0.204i)12-s − 0.581i·13-s + (0.742 + 0.742i)14-s + (−0.417 − 0.399i)15-s + 0.250·16-s − 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.401 + 0.915i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.401 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760774591\)
\(L(\frac12)\) \(\approx\) \(1.760774591\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0484 + 2.23i)T \)
37 \( 1 + (5.20 + 3.14i)T \)
good7 \( 1 + (-2.77 + 2.77i)T - 7iT^{2} \)
11 \( 1 - 2.74iT - 11T^{2} \)
13 \( 1 + 2.09iT - 13T^{2} \)
17 \( 1 + 4.54T + 17T^{2} \)
19 \( 1 + (-3.00 + 3.00i)T - 19iT^{2} \)
23 \( 1 + 1.09iT - 23T^{2} \)
29 \( 1 + (-3.41 - 3.41i)T + 29iT^{2} \)
31 \( 1 + (0.444 - 0.444i)T - 31iT^{2} \)
41 \( 1 + 3.49iT - 41T^{2} \)
43 \( 1 + 2.64iT - 43T^{2} \)
47 \( 1 + (4.28 - 4.28i)T - 47iT^{2} \)
53 \( 1 + (4.10 + 4.10i)T + 53iT^{2} \)
59 \( 1 + (-10.1 + 10.1i)T - 59iT^{2} \)
61 \( 1 + (4.77 - 4.77i)T - 61iT^{2} \)
67 \( 1 + (-3.42 - 3.42i)T + 67iT^{2} \)
71 \( 1 - 0.512T + 71T^{2} \)
73 \( 1 + (-9.57 + 9.57i)T - 73iT^{2} \)
79 \( 1 + (-4.84 + 4.84i)T - 79iT^{2} \)
83 \( 1 + (-4.09 - 4.09i)T + 83iT^{2} \)
89 \( 1 + (-9.59 - 9.59i)T + 89iT^{2} \)
97 \( 1 + 5.08T + 97T^{2} \)
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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

−9.402944201370828454548871894828, −8.675484582724881655231479697841, −7.965428049257276292926610593417, −7.33599327764015257359506595661, −6.59066806631861893769868496700, −5.13116456255341929338711782504, −4.74341850424701016302711901074, −3.72958483000790235256100463867, −1.96749109700020793448701111588, −0.76120191624696638823390899751, 1.81380962958062260532610592036, 2.66831810899463373947529290297, 3.59570717467194960459924651199, 4.64588327876056614960252575454, 5.60117408940082457386326504317, 6.57005976054208704373405593391, 7.84735591191128233989410458172, 8.479726115457472355221569904776, 9.215815947247531649109654780081, 10.06904416466150610747869333090

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