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Properties

Label 2-1122-17.13-c1-0-24
Degree $2$
Conductor $1122$
Sign $0.880 + 0.473i$
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 − i)5-s + (−0.707 + 0.707i)6-s + (2.65 − 2.65i)7-s − i·8-s + 1.00i·9-s + (1 − i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s − 4.33·13-s + (2.65 + 2.65i)14-s − 1.41i·15-s + 16-s + (−4.06 + 0.694i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.447 − 0.447i)5-s + (−0.288 + 0.288i)6-s + (1.00 − 1.00i)7-s − 0.353i·8-s + 0.333i·9-s + (0.316 − 0.316i)10-s + (0.213 − 0.213i)11-s + (−0.204 − 0.204i)12-s − 1.20·13-s + (0.710 + 0.710i)14-s − 0.365i·15-s + 0.250·16-s + (−0.985 + 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $0.880 + 0.473i$
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: \((2,\ 1122,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.469290268\)
\(L(\frac12)\) \(\approx\) \(1.469290268\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.06 - 0.694i)T \)
good5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 + (-2.65 + 2.65i)T - 7iT^{2} \)
13 \( 1 + 4.33T + 13T^{2} \)
19 \( 1 + 3.75iT - 19T^{2} \)
23 \( 1 + (-6.41 + 6.41i)T - 23iT^{2} \)
29 \( 1 + (-4.35 - 4.35i)T + 29iT^{2} \)
31 \( 1 + (4.79 + 4.79i)T + 31iT^{2} \)
37 \( 1 + (1.40 + 1.40i)T + 37iT^{2} \)
41 \( 1 + (-3.77 + 3.77i)T - 41iT^{2} \)
43 \( 1 + 7.68iT - 43T^{2} \)
47 \( 1 + 2.53T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + (1.57 - 1.57i)T - 61iT^{2} \)
67 \( 1 - 8.16T + 67T^{2} \)
71 \( 1 + (-6.99 - 6.99i)T + 71iT^{2} \)
73 \( 1 + (-5.91 - 5.91i)T + 73iT^{2} \)
79 \( 1 + (-7.39 + 7.39i)T - 79iT^{2} \)
83 \( 1 - 7.35iT - 83T^{2} \)
89 \( 1 + 0.812T + 89T^{2} \)
97 \( 1 + (1.98 + 1.98i)T + 97iT^{2} \)
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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

−9.509113965850766960922997838817, −8.763290617010025822659164275295, −8.177770668334524320851057863765, −7.27405218055058912519017129175, −6.74159392709434349904684678889, −5.13543832404214276621096212108, −4.63345705333634954577384174231, −3.97255917019799487974728260757, −2.44255410329703620566269500166, −0.63252642590093753398735041001, 1.57644539263844105105941734200, 2.47880198190255369023131247530, 3.41347527343710603554532379446, 4.67423438798024663288227177646, 5.38206480472728870709387187505, 6.72744352089237014049315203726, 7.60128264067871507136474595813, 8.274104300562940378819874528749, 9.167974012405768259343650752145, 9.743690529927542364676564316122

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