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Properties

Label 2-5520-92.91-c1-0-50
Degree $2$
Conductor $5520$
Sign $0.720 - 0.693i$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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  + i·3-s − i·5-s + 0.143·7-s − 9-s + 4.80·11-s + 4.78·13-s + 15-s + 4.86i·17-s + 0.379·19-s + 0.143i·21-s + (1.15 + 4.65i)23-s − 25-s − i·27-s + 8.39·29-s − 2.55i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s + 0.0540·7-s − 0.333·9-s + 1.44·11-s + 1.32·13-s + 0.258·15-s + 1.18i·17-s + 0.0870·19-s + 0.0312i·21-s + (0.240 + 0.970i)23-s − 0.200·25-s − 0.192i·27-s + 1.55·29-s − 0.459i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.423803128\)
\(L(\frac12)\) \(\approx\) \(2.423803128\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + iT \)
23 \( 1 + (-1.15 - 4.65i)T \)
good7 \( 1 - 0.143T + 7T^{2} \)
11 \( 1 - 4.80T + 11T^{2} \)
13 \( 1 - 4.78T + 13T^{2} \)
17 \( 1 - 4.86iT - 17T^{2} \)
19 \( 1 - 0.379T + 19T^{2} \)
29 \( 1 - 8.39T + 29T^{2} \)
31 \( 1 + 2.55iT - 31T^{2} \)
37 \( 1 + 5.56iT - 37T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 - 8.93iT - 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 - 6.57iT - 59T^{2} \)
61 \( 1 - 7.49iT - 61T^{2} \)
67 \( 1 - 4.96T + 67T^{2} \)
71 \( 1 + 6.77iT - 71T^{2} \)
73 \( 1 + 8.04T + 73T^{2} \)
79 \( 1 + 1.53T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 + 12.6iT - 89T^{2} \)
97 \( 1 - 5.67iT - 97T^{2} \)
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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

−8.461911487794653110788036253299, −7.68007610220928471814480603478, −6.53689381608694893752944707534, −6.17927055135755850030977483923, −5.35735496063580905358824304882, −4.41257153511814223840371717919, −3.85249960334417314776496555277, −3.23658821207426479848924255521, −1.77961797246990955489277272915, −1.03120039176955039430056847091, 0.802249811835910226742798288660, 1.60134127424196461449661110591, 2.77740899619611433565179510622, 3.44571337554512173819710068033, 4.36311625419157854043470283803, 5.19301464675046585189096076003, 6.31539432693125606082497947643, 6.55768824515997623760005243700, 7.13838322968422536184890196444, 8.200649678095559161240651481965

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