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Properties

Label 2-1122-17.13-c1-0-19
Degree $2$
Conductor $1122$
Sign $0.987 - 0.158i$
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 + (1.77 + 1.77i)5-s + (0.707 − 0.707i)6-s + (2.79 − 2.79i)7-s − i·8-s + 1.00i·9-s + (−1.77 + 1.77i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s − 1.10·13-s + (2.79 + 2.79i)14-s − 2.51i·15-s + 16-s + (2.80 − 3.02i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.793 + 0.793i)5-s + (0.288 − 0.288i)6-s + (1.05 − 1.05i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.561 + 0.561i)10-s + (0.213 − 0.213i)11-s + (0.204 + 0.204i)12-s − 0.306·13-s + (0.746 + 0.746i)14-s − 0.648i·15-s + 0.250·16-s + (0.680 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $0.987 - 0.158i$
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: \((2,\ 1122,\ (\ :1/2),\ 0.987 - 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.747639732\)
\(L(\frac12)\) \(\approx\) \(1.747639732\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.80 + 3.02i)T \)
good5 \( 1 + (-1.77 - 1.77i)T + 5iT^{2} \)
7 \( 1 + (-2.79 + 2.79i)T - 7iT^{2} \)
13 \( 1 + 1.10T + 13T^{2} \)
19 \( 1 - 1.82iT - 19T^{2} \)
23 \( 1 + (-2.72 + 2.72i)T - 23iT^{2} \)
29 \( 1 + (3.36 + 3.36i)T + 29iT^{2} \)
31 \( 1 + (0.842 + 0.842i)T + 31iT^{2} \)
37 \( 1 + (5.62 + 5.62i)T + 37iT^{2} \)
41 \( 1 + (-3.84 + 3.84i)T - 41iT^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + 4.91iT - 53T^{2} \)
59 \( 1 - 5.33iT - 59T^{2} \)
61 \( 1 + (4.95 - 4.95i)T - 61iT^{2} \)
67 \( 1 - 9.35T + 67T^{2} \)
71 \( 1 + (1.65 + 1.65i)T + 71iT^{2} \)
73 \( 1 + (-7.10 - 7.10i)T + 73iT^{2} \)
79 \( 1 + (-3.86 + 3.86i)T - 79iT^{2} \)
83 \( 1 - 7.40iT - 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (1.28 + 1.28i)T + 97iT^{2} \)
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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

−9.941748852416188986497061565559, −8.968541595756331477120271924596, −7.77636893270585516171900298191, −7.39432070700923022657460568611, −6.56517345443690986498866257465, −5.73241056420965018614755968181, −4.91743887337115461772566746966, −3.85243809909573720387213869968, −2.35385601537468379634945538466, −0.966143201816559379699658769408, 1.32096641491345926422659967751, 2.18009735045041343749716208495, 3.59245360624992415106932013106, 4.91308180622565361898087072053, 5.21580949457417454038583405401, 6.02758553929226903946467596138, 7.47498217474308979281964022072, 8.629896588239981696354008219993, 9.045567919638162651802774388799, 9.780570302978225565931317556087

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