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Properties

Label 2-1190-119.16-c1-0-18
Degree $2$
Conductor $1190$
Sign $0.139 - 0.990i$
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 + (1.29 − 0.748i)3-s + (−0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s + 1.49i·6-s + (2.63 + 0.195i)7-s + 0.999·8-s + (−0.379 + 0.658i)9-s + (0.866 − 0.499i)10-s + (−4.84 + 2.79i)11-s + (−1.29 − 0.748i)12-s + 0.757·13-s + (−1.48 + 2.18i)14-s − 1.49·15-s + (−0.5 + 0.866i)16-s + (3.80 − 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.748 − 0.432i)3-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + 0.611i·6-s + (0.997 + 0.0740i)7-s + 0.353·8-s + (−0.126 + 0.219i)9-s + (0.273 − 0.158i)10-s + (−1.46 + 0.843i)11-s + (−0.374 − 0.216i)12-s + 0.210·13-s + (−0.397 + 0.584i)14-s − 0.386·15-s + (−0.125 + 0.216i)16-s + (0.923 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.139 - 0.990i$
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: \((2,\ 1190,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487245813\)
\(L(\frac12)\) \(\approx\) \(1.487245813\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.195i)T \)
17 \( 1 + (-3.80 + 1.58i)T \)
good3 \( 1 + (-1.29 + 0.748i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (4.84 - 2.79i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.757T + 13T^{2} \)
19 \( 1 + (4.08 - 7.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.94 - 4.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.97iT - 29T^{2} \)
31 \( 1 + (4.71 - 2.72i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.54 - 4.93i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 + 1.98T + 43T^{2} \)
47 \( 1 + (-6.18 + 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.26 + 3.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.97 + 3.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.0 - 5.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.69 - 6.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (7.07 - 4.08i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.57 - 3.79i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.31T + 83T^{2} \)
89 \( 1 + (3.41 - 5.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.62iT - 97T^{2} \)
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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

−9.863739533273983092021205120281, −8.631938611112441579353878869049, −8.326242196600954644646483564624, −7.50419434440623505131059973570, −7.19855662614451878495614742348, −5.43283430645761212700681965684, −5.20310030482179872020937008782, −3.85716341025864333274550610694, −2.49103967257161773852699412390, −1.45347138082945158183109450227, 0.69314163408059535303053972019, 2.51649620817875535425618320684, 3.03700721880967842954123418596, 4.19863632454114619853205254048, 4.99254226255576851993584430845, 6.21569193991213719022895080403, 7.62834560569292822526157398110, 8.056558444119133922187674353959, 8.804982214695089991403468165761, 9.425192422669274136300191460273

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