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Properties

Label 2-1190-17.4-c1-0-10
Degree $2$
Conductor $1190$
Sign $0.834 - 0.550i$
Analytic cond. $9.50219$
Root an. cond. $3.08256$
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 + (−2.16 + 2.16i)3-s − 4-s + (−0.707 + 0.707i)5-s + (2.16 + 2.16i)6-s + (0.707 + 0.707i)7-s + i·8-s − 6.34i·9-s + (0.707 + 0.707i)10-s + (−1.53 − 1.53i)11-s + (2.16 − 2.16i)12-s + 2.66·13-s + (0.707 − 0.707i)14-s − 3.05i·15-s + 16-s + (4.11 + 0.328i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.24 + 1.24i)3-s − 0.5·4-s + (−0.316 + 0.316i)5-s + (0.882 + 0.882i)6-s + (0.267 + 0.267i)7-s + 0.353i·8-s − 2.11i·9-s + (0.223 + 0.223i)10-s + (−0.461 − 0.461i)11-s + (0.623 − 0.623i)12-s + 0.739·13-s + (0.188 − 0.188i)14-s − 0.789i·15-s + 0.250·16-s + (0.996 + 0.0796i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8741744464\)
\(L(\frac12)\) \(\approx\) \(0.8741744464\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-4.11 - 0.328i)T \)
good3 \( 1 + (2.16 - 2.16i)T - 3iT^{2} \)
11 \( 1 + (1.53 + 1.53i)T + 11iT^{2} \)
13 \( 1 - 2.66T + 13T^{2} \)
19 \( 1 + 5.16iT - 19T^{2} \)
23 \( 1 + (0.0971 + 0.0971i)T + 23iT^{2} \)
29 \( 1 + (-1.11 + 1.11i)T - 29iT^{2} \)
31 \( 1 + (3.24 - 3.24i)T - 31iT^{2} \)
37 \( 1 + (0.866 - 0.866i)T - 37iT^{2} \)
41 \( 1 + (-5.42 - 5.42i)T + 41iT^{2} \)
43 \( 1 + 4.69iT - 43T^{2} \)
47 \( 1 - 6.20T + 47T^{2} \)
53 \( 1 - 7.47iT - 53T^{2} \)
59 \( 1 - 8.44iT - 59T^{2} \)
61 \( 1 + (-6.20 - 6.20i)T + 61iT^{2} \)
67 \( 1 - 4.45T + 67T^{2} \)
71 \( 1 + (10.5 - 10.5i)T - 71iT^{2} \)
73 \( 1 + (-2.77 + 2.77i)T - 73iT^{2} \)
79 \( 1 + (-0.833 - 0.833i)T + 79iT^{2} \)
83 \( 1 + 6.41iT - 83T^{2} \)
89 \( 1 - 9.45T + 89T^{2} \)
97 \( 1 + (-5.65 + 5.65i)T - 97iT^{2} \)
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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

−10.22061825558621856456822992316, −9.232304983312944227550966896617, −8.563628072971590427528860992000, −7.32804124353841205300108832667, −6.07238198920812611954192670645, −5.46981905946437228530165518884, −4.61658959122008842913638830480, −3.78458425403630448101343020708, −2.85659272350803884172035839052, −0.850771094259495361509822674396, 0.68349536231519790246994432730, 1.81146127166060796546459078939, 3.75777649964176657373603190144, 4.95549281953518466341225420112, 5.64371189061333479975048105044, 6.29277850394958882156313398400, 7.28004172119733216329951190211, 7.75004551132835423815474476033, 8.405986487991469345948018839141, 9.719968676119581983532037061888

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