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Properties

Label 2-1122-33.32-c1-0-50
Degree $2$
Conductor $1122$
Sign $-0.536 + 0.843i$
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  − 2-s + (1.55 − 0.758i)3-s + 4-s − 0.724i·5-s + (−1.55 + 0.758i)6-s − 3.18i·7-s − 8-s + (1.84 − 2.36i)9-s + 0.724i·10-s + (−2.82 + 1.73i)11-s + (1.55 − 0.758i)12-s − 1.44i·13-s + 3.18i·14-s + (−0.549 − 1.12i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.899 − 0.437i)3-s + 0.5·4-s − 0.323i·5-s + (−0.635 + 0.309i)6-s − 1.20i·7-s − 0.353·8-s + (0.616 − 0.787i)9-s + 0.229i·10-s + (−0.852 + 0.523i)11-s + (0.449 − 0.218i)12-s − 0.399i·13-s + 0.851i·14-s + (−0.141 − 0.291i)15-s + 0.250·16-s + 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.536 + 0.843i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -0.536 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.319334404\)
\(L(\frac12)\) \(\approx\) \(1.319334404\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.55 + 0.758i)T \)
11 \( 1 + (2.82 - 1.73i)T \)
17 \( 1 - T \)
good5 \( 1 + 0.724iT - 5T^{2} \)
7 \( 1 + 3.18iT - 7T^{2} \)
13 \( 1 + 1.44iT - 13T^{2} \)
19 \( 1 + 1.10iT - 19T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + 3.14T + 29T^{2} \)
31 \( 1 + 6.80T + 31T^{2} \)
37 \( 1 + 5.95T + 37T^{2} \)
41 \( 1 - 3.35T + 41T^{2} \)
43 \( 1 - 1.64iT - 43T^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 - 1.92iT - 53T^{2} \)
59 \( 1 - 5.57iT - 59T^{2} \)
61 \( 1 - 5.40iT - 61T^{2} \)
67 \( 1 - 0.828T + 67T^{2} \)
71 \( 1 + 9.17iT - 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 7.66T + 83T^{2} \)
89 \( 1 - 3.79iT - 89T^{2} \)
97 \( 1 - 3.88T + 97T^{2} \)
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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

−9.405654730836030707665864781002, −8.724913701670225797225060704972, −7.82546048506533859320896562498, −7.35872285919600863823705762507, −6.64235727150336411498089332708, −5.24560761626271899209160078142, −4.08485843615284455890716483485, −3.05105582150847881947354489544, −1.89449136678425815157300386408, −0.63322765766291631513216323301, 1.85616345752060661760616360175, 2.76465152036404607698277001973, 3.58795511892471546071135056093, 5.08667037728470555935514645661, 5.85376659892678569518505572643, 7.09499258004723633919070340586, 7.85339527005229122194284322058, 8.642993251217862949408421111010, 9.168430864353053114429235760075, 9.897152892743367269837312189958

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