Properties

Label 2-1170-65.4-c1-0-24
Degree $2$
Conductor $1170$
Sign $0.957 + 0.289i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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 + (−0.499 − 0.866i)4-s + (2.03 − 0.928i)5-s + (−1.40 − 2.42i)7-s + 0.999·8-s + (−0.213 + 2.22i)10-s + (0.515 + 0.297i)11-s + (1.10 + 3.43i)13-s + 2.80·14-s + (−0.5 + 0.866i)16-s + (4.98 − 2.87i)17-s + (−6.59 + 3.80i)19-s + (−1.82 − 1.29i)20-s + (−0.515 + 0.297i)22-s + (4.02 + 2.32i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.909 − 0.415i)5-s + (−0.530 − 0.918i)7-s + 0.353·8-s + (−0.0673 + 0.703i)10-s + (0.155 + 0.0896i)11-s + (0.307 + 0.951i)13-s + 0.749·14-s + (−0.125 + 0.216i)16-s + (1.20 − 0.697i)17-s + (−1.51 + 0.873i)19-s + (−0.407 − 0.290i)20-s + (−0.109 + 0.0634i)22-s + (0.838 + 0.484i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.957 + 0.289i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.480751564\)
\(L(\frac12)\) \(\approx\) \(1.480751564\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.03 + 0.928i)T \)
13 \( 1 + (-1.10 - 3.43i)T \)
good7 \( 1 + (1.40 + 2.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.515 - 0.297i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-4.98 + 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.59 - 3.80i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.02 - 2.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 + (-5.18 + 8.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 2.87i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.67 + 2.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 + (8.40 - 4.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 + 6.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.11 + 0.642i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 - 4.19T + 83T^{2} \)
89 \( 1 + (-5.24 - 3.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.45 - 14.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.537355059008307237363778260695, −9.144702217368558178206582181952, −8.030137720495486505898494597421, −7.24640997539215001912882713119, −6.35061120782310967901823757889, −5.79893046263675921639487731515, −4.63177247703167859383488779207, −3.76567048411330762544412411307, −2.12498331431830334937418341172, −0.834356762703838226555041018965, 1.27090327490422101719201129700, 2.67359173218176866352013722905, 3.10471430992265145214096060231, 4.61177218232192961897195175304, 5.77994722341485865429051656080, 6.28995485349442912545222010313, 7.39581272895196647145854122551, 8.613446198040703643040934426162, 8.949789730414172753753437500936, 9.946564957902786757655972139481

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