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Properties

Label 2-1110-185.142-c1-0-7
Degree $2$
Conductor $1110$
Sign $0.819 - 0.573i$
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 + (−2.17 + 0.530i)5-s + (0.707 − 0.707i)6-s + (−1.93 − 1.93i)7-s + i·8-s + 1.00i·9-s + (0.530 + 2.17i)10-s − 2.54i·11-s + (−0.707 − 0.707i)12-s + 2.64i·13-s + (−1.93 + 1.93i)14-s + (−1.91 − 1.16i)15-s + 16-s + 3.60·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.971 + 0.237i)5-s + (0.288 − 0.288i)6-s + (−0.730 − 0.730i)7-s + 0.353i·8-s + 0.333i·9-s + (0.167 + 0.686i)10-s − 0.766i·11-s + (−0.204 − 0.204i)12-s + 0.732i·13-s + (−0.516 + 0.516i)14-s + (−0.493 − 0.299i)15-s + 0.250·16-s + 0.874·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.819 - 0.573i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.104483641\)
\(L(\frac12)\) \(\approx\) \(1.104483641\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 0.530i)T \)
37 \( 1 + (1.59 - 5.87i)T \)
good7 \( 1 + (1.93 + 1.93i)T + 7iT^{2} \)
11 \( 1 + 2.54iT - 11T^{2} \)
13 \( 1 - 2.64iT - 13T^{2} \)
17 \( 1 - 3.60T + 17T^{2} \)
19 \( 1 + (-2.47 - 2.47i)T + 19iT^{2} \)
23 \( 1 - 4.49iT - 23T^{2} \)
29 \( 1 + (4.58 - 4.58i)T - 29iT^{2} \)
31 \( 1 + (-7.33 - 7.33i)T + 31iT^{2} \)
41 \( 1 + 4.31iT - 41T^{2} \)
43 \( 1 - 3.12iT - 43T^{2} \)
47 \( 1 + (-4.51 - 4.51i)T + 47iT^{2} \)
53 \( 1 + (0.301 - 0.301i)T - 53iT^{2} \)
59 \( 1 + (-0.813 - 0.813i)T + 59iT^{2} \)
61 \( 1 + (0.867 + 0.867i)T + 61iT^{2} \)
67 \( 1 + (5.77 - 5.77i)T - 67iT^{2} \)
71 \( 1 + 2.30T + 71T^{2} \)
73 \( 1 + (-3.70 - 3.70i)T + 73iT^{2} \)
79 \( 1 + (-4.70 - 4.70i)T + 79iT^{2} \)
83 \( 1 + (0.524 - 0.524i)T - 83iT^{2} \)
89 \( 1 + (6.69 - 6.69i)T - 89iT^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−10.05463448889283687510763724006, −9.230212666364959446719903423494, −8.391591524230820349118448652396, −7.58082388289025487110195537758, −6.74754798328181428572013624163, −5.45390439796690000410787320005, −4.32040788602255958587218236356, −3.47373020113267931554773661349, −3.09623396273592612128167504741, −1.19937697826369601965326326052, 0.54374969018397289561139584174, 2.53868523576464610137771650553, 3.55536423532633559659551255829, 4.59415885416607206716140180586, 5.63879085672313021118532787585, 6.49425183645858055101039768673, 7.52230452789889123064474487771, 7.86325516762031301902483320153, 8.830182880942106359988980896896, 9.489115672918530171823384375856

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