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Properties

Label 2-1122-17.13-c1-0-13
Degree $2$
Conductor $1122$
Sign $0.788 - 0.615i$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
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 + (2.41 + 2.41i)5-s + (0.707 − 0.707i)6-s + i·8-s + 1.00i·9-s + (2.41 − 2.41i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + 2.82·13-s + 3.41i·15-s + 16-s + (−1 + 4i)17-s + 1.00·18-s − 0.828i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (1.07 + 1.07i)5-s + (0.288 − 0.288i)6-s + 0.353i·8-s + 0.333i·9-s + (0.763 − 0.763i)10-s + (−0.213 + 0.213i)11-s + (−0.204 − 0.204i)12-s + 0.784·13-s + 0.881i·15-s + 0.250·16-s + (−0.242 + 0.970i)17-s + 0.235·18-s − 0.190i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ 0.788 - 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.092163192\)
\(L(\frac12)\) \(\approx\) \(2.092163192\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (1 - 4i)T \)
good5 \( 1 + (-2.41 - 2.41i)T + 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (-0.828 + 0.828i)T - 23iT^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + (-2 - 2i)T + 31iT^{2} \)
37 \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \)
41 \( 1 + (5.82 - 5.82i)T - 41iT^{2} \)
43 \( 1 + 4.82iT - 43T^{2} \)
47 \( 1 - 8.82T + 47T^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 - 0.828iT - 59T^{2} \)
61 \( 1 + (1.58 - 1.58i)T - 61iT^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + (-5.65 - 5.65i)T + 71iT^{2} \)
73 \( 1 + (4.07 + 4.07i)T + 73iT^{2} \)
79 \( 1 + (-8.82 + 8.82i)T - 79iT^{2} \)
83 \( 1 + 9.65iT - 83T^{2} \)
89 \( 1 - 9.17T + 89T^{2} \)
97 \( 1 + (1 + i)T + 97iT^{2} \)
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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

−10.12671662686389993533627888316, −9.272168106491016101504912778651, −8.571823867540413478572036754095, −7.50637772041663799276820418990, −6.39644220134895740768221763664, −5.72696869585987445678323968017, −4.50013709268668293122501749901, −3.47057471870926818417574071751, −2.61318338975039817773870970932, −1.68898516530311907867568628175, 0.928631664912966257754389363238, 2.14966229271506686751957181613, 3.58767000637891339174228522269, 4.86070657285716384491230213781, 5.54086418008100514368051642594, 6.33698559939816904732372530848, 7.24746381529542491808345652237, 8.221756821614395793691860248230, 8.882750834296987774179392726442, 9.391599515315582539829544033073

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