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Properties

Label 2-1620-45.34-c1-0-22
Degree $2$
Conductor $1620$
Sign $0.298 + 0.954i$
Analytic cond. $12.9357$
Root an. cond. $3.59663$
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  + (2.23 + 0.103i)5-s + (−1.10 − 0.640i)7-s + (2.07 − 3.58i)11-s + (5.64 − 3.26i)13-s − 5.98i·17-s − 7.17·19-s + (−6.52 + 3.76i)23-s + (4.97 + 0.461i)25-s + (−2.59 + 4.5i)29-s + (−2.58 − 4.48i)31-s + (−2.41 − 1.54i)35-s − 5.24i·37-s + (0.340 + 0.589i)41-s + (1.10 + 0.640i)43-s + (4.59 + 2.65i)47-s + ⋯
L(s)  = 1  + (0.998 + 0.0462i)5-s + (−0.419 − 0.242i)7-s + (0.624 − 1.08i)11-s + (1.56 − 0.904i)13-s − 1.45i·17-s − 1.64·19-s + (−1.35 + 0.785i)23-s + (0.995 + 0.0923i)25-s + (−0.482 + 0.835i)29-s + (−0.465 − 0.805i)31-s + (−0.407 − 0.261i)35-s − 0.861i·37-s + (0.0531 + 0.0920i)41-s + (0.169 + 0.0977i)43-s + (0.670 + 0.387i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.298 + 0.954i$
Analytic conductor: \(12.9357\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1/2),\ 0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.886002042\)
\(L(\frac12)\) \(\approx\) \(1.886002042\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.23 - 0.103i)T \)
good7 \( 1 + (1.10 + 0.640i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.07 + 3.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.64 + 3.26i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.98iT - 17T^{2} \)
19 \( 1 + 7.17T + 19T^{2} \)
23 \( 1 + (6.52 - 3.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 - 4.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.58 + 4.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.24iT - 37T^{2} \)
41 \( 1 + (-0.340 - 0.589i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.10 - 0.640i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.59 - 2.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.21iT - 53T^{2} \)
59 \( 1 + (-3.80 - 6.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.08 + 1.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.50T + 71T^{2} \)
73 \( 1 + 7.80iT - 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.44 - 4.87i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-12.4 - 7.16i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.164405228794698668490751448476, −8.650649918408773397451673578220, −7.67579178151615547681576869094, −6.55311458693297112147428434050, −6.02326837515154267448521340552, −5.40680378348403521235371490167, −3.98469989422692200597202715772, −3.26357669605477319466377422091, −2.04386572690435973831271404783, −0.72689406173775826664261221204, 1.62787322620764174368517066621, 2.19362699795057648520404261815, 3.84231218891319328634521973959, 4.34497524976520158018753995564, 5.75319132288522652880803185131, 6.39909692165787782041483116080, 6.72740557079575702773365106444, 8.293827705831403080743583623965, 8.725761120163765051896065074816, 9.573511457841718287508163180119

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