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Properties

Label 2-1320-88.43-c1-0-17
Degree $2$
Conductor $1320$
Sign $0.242 - 0.970i$
Analytic cond. $10.5402$
Root an. cond. $3.24657$
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  + (−1.19 − 0.762i)2-s + 3-s + (0.837 + 1.81i)4-s + i·5-s + (−1.19 − 0.762i)6-s − 2.25·7-s + (0.387 − 2.80i)8-s + 9-s + (0.762 − 1.19i)10-s + (0.357 − 3.29i)11-s + (0.837 + 1.81i)12-s − 1.99·13-s + (2.68 + 1.72i)14-s + i·15-s + (−2.59 + 3.04i)16-s + 6.18i·17-s + ⋯
L(s)  = 1  + (−0.842 − 0.539i)2-s + 0.577·3-s + (0.418 + 0.908i)4-s + 0.447i·5-s + (−0.486 − 0.311i)6-s − 0.853·7-s + (0.136 − 0.990i)8-s + 0.333·9-s + (0.241 − 0.376i)10-s + (0.107 − 0.994i)11-s + (0.241 + 0.524i)12-s − 0.553·13-s + (0.718 + 0.459i)14-s + 0.258i·15-s + (−0.649 + 0.760i)16-s + 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1320\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 11\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(10.5402\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1320} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1320,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8444550445\)
\(L(\frac12)\) \(\approx\) \(0.8444550445\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.762i)T \)
3 \( 1 - T \)
5 \( 1 - iT \)
11 \( 1 + (-0.357 + 3.29i)T \)
good7 \( 1 + 2.25T + 7T^{2} \)
13 \( 1 + 1.99T + 13T^{2} \)
17 \( 1 - 6.18iT - 17T^{2} \)
19 \( 1 - 2.26iT - 19T^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 - 5.98T + 29T^{2} \)
31 \( 1 + 4.92iT - 31T^{2} \)
37 \( 1 - 4.55iT - 37T^{2} \)
41 \( 1 - 5.53iT - 41T^{2} \)
43 \( 1 - 3.59iT - 43T^{2} \)
47 \( 1 - 6.66iT - 47T^{2} \)
53 \( 1 - 8.98iT - 53T^{2} \)
59 \( 1 + 4.04T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 - 4.33T + 67T^{2} \)
71 \( 1 + 1.59iT - 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 - 0.723T + 79T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 6.02T + 97T^{2} \)
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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

−9.759888436784096653182845046894, −9.145881001404405329460042950855, −8.163325028059623195774223808447, −7.76128627823037213711815178023, −6.56625687988972686469364643509, −6.05745439883933837155810688036, −4.25182347527747883079518259039, −3.33519365467211788935587989293, −2.74107437466648524138963103528, −1.38670317095941804422513970610, 0.43486298304694246452518127791, 2.03760121383532001110099927032, 3.00350751172173617299830726498, 4.56642896319019684773305349059, 5.23332271150005449681301632090, 6.59387317865240604243852322582, 7.05267884779055569154579507584, 7.82608845673537881223977977286, 8.875316694214121242883179033016, 9.281244549888896833075941194038

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