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Properties

Label 2-1122-17.13-c1-0-15
Degree $2$
Conductor $1122$
Sign $-0.447 - 0.894i$
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 + (2.53 + 2.53i)5-s + (−0.707 + 0.707i)6-s + (1.12 − 1.12i)7-s − i·8-s + 1.00i·9-s + (−2.53 + 2.53i)10-s + (−0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + 5.39·13-s + (1.12 + 1.12i)14-s + 3.58i·15-s + 16-s + (−3.72 − 1.76i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (1.13 + 1.13i)5-s + (−0.288 + 0.288i)6-s + (0.426 − 0.426i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.800 + 0.800i)10-s + (−0.213 + 0.213i)11-s + (−0.204 − 0.204i)12-s + 1.49·13-s + (0.301 + 0.301i)14-s + 0.924i·15-s + 0.250·16-s + (−0.903 − 0.429i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 1122,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.320970048\)
\(L(\frac12)\) \(\approx\) \(2.320970048\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.72 + 1.76i)T \)
good5 \( 1 + (-2.53 - 2.53i)T + 5iT^{2} \)
7 \( 1 + (-1.12 + 1.12i)T - 7iT^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
19 \( 1 - 6.49iT - 19T^{2} \)
23 \( 1 + (-4.90 + 4.90i)T - 23iT^{2} \)
29 \( 1 + (-0.546 - 0.546i)T + 29iT^{2} \)
31 \( 1 + (3.25 + 3.25i)T + 31iT^{2} \)
37 \( 1 + (0.308 + 0.308i)T + 37iT^{2} \)
41 \( 1 + (-4.80 + 4.80i)T - 41iT^{2} \)
43 \( 1 - 12.3iT - 43T^{2} \)
47 \( 1 + 9.18T + 47T^{2} \)
53 \( 1 + 4.69iT - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 + (-4.64 + 4.64i)T - 61iT^{2} \)
67 \( 1 + 5.47T + 67T^{2} \)
71 \( 1 + (11.3 + 11.3i)T + 71iT^{2} \)
73 \( 1 + (0.0688 + 0.0688i)T + 73iT^{2} \)
79 \( 1 + (5.29 - 5.29i)T - 79iT^{2} \)
83 \( 1 - 9.69iT - 83T^{2} \)
89 \( 1 - 7.97T + 89T^{2} \)
97 \( 1 + (-0.931 - 0.931i)T + 97iT^{2} \)
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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

−9.985639674066065283289809327119, −9.301660288983788591359112245514, −8.407323813056697322059629298620, −7.61445240184577131650421430249, −6.57170502698261029376282787852, −6.13284073439963987371387259941, −5.05139265877610418472404497290, −3.98805064302434475610032266060, −2.96555049323178932453995577815, −1.70516024383555692153925782016, 1.10535989768651222125641316603, 1.89021375513969348269896841955, 2.98452454102790553879122963223, 4.30558692334932646919176649418, 5.26971191550754785704590325335, 5.93809389217981610814002554709, 7.07971805659028572370111146181, 8.475987208953787327817736941058, 8.843975522080878585294349088112, 9.245723383669974536561014420455

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