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Properties

Label 2-1190-119.16-c1-0-1
Degree $2$
Conductor $1190$
Sign $-0.492 - 0.870i$
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  + (0.5 − 0.866i)2-s + (−0.797 + 0.460i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + 0.920i·6-s + (0.0419 − 2.64i)7-s − 0.999·8-s + (−1.07 + 1.86i)9-s + (0.866 − 0.499i)10-s + (−0.406 + 0.234i)11-s + (0.797 + 0.460i)12-s − 6.11·13-s + (−2.27 − 1.35i)14-s − 0.920·15-s + (−0.5 + 0.866i)16-s + (3.44 + 2.26i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.460 + 0.265i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 0.375i·6-s + (0.0158 − 0.999i)7-s − 0.353·8-s + (−0.358 + 0.621i)9-s + (0.273 − 0.158i)10-s + (−0.122 + 0.0707i)11-s + (0.230 + 0.132i)12-s − 1.69·13-s + (−0.606 − 0.363i)14-s − 0.237·15-s + (−0.125 + 0.216i)16-s + (0.836 + 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ -0.492 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2948023155\)
\(L(\frac12)\) \(\approx\) \(0.2948023155\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.0419 + 2.64i)T \)
17 \( 1 + (-3.44 - 2.26i)T \)
good3 \( 1 + (0.797 - 0.460i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.406 - 0.234i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.11T + 13T^{2} \)
19 \( 1 + (2.14 - 3.71i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.86 + 3.38i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.33iT - 29T^{2} \)
31 \( 1 + (1.17 - 0.680i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.05 - 1.18i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.14iT - 41T^{2} \)
43 \( 1 + 6.74T + 43T^{2} \)
47 \( 1 + (3.40 - 5.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.57 + 2.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.167 + 0.290i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.23 - 1.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 - 2.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.44iT - 71T^{2} \)
73 \( 1 + (9.02 - 5.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.1 + 6.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.25T + 83T^{2} \)
89 \( 1 + (8.57 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.37iT - 97T^{2} \)
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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

−10.10577046659691546442011420885, −9.819337735397381263072212507323, −8.317669773779553693763912161841, −7.60512434302201766632563400876, −6.54498593367744201081107364794, −5.61605063736604373651365970258, −4.82608207733140225006049714799, −4.02581819057438872553321994774, −2.81616041316738383365281853632, −1.72438078188395757091564503161, 0.11004928599745489978186610126, 2.16521296157738051768193436133, 3.19408932940991878874565944790, 4.65312851970339888143959291785, 5.42359723723620428378063120325, 5.96720692808366344845870090862, 6.87812551890025652940069111479, 7.70304981156403451377442061502, 8.632053950534602216006103889588, 9.474611097545561341642529341627

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