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Properties

Label 2-920-5.4-c1-0-7
Degree $2$
Conductor $920$
Sign $0.487 - 0.872i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 0.493i·3-s + (1.09 − 1.95i)5-s + 4.54i·7-s + 2.75·9-s − 4.61·11-s + 5.54i·13-s + (−0.963 − 0.538i)15-s + 7.16i·17-s − 1.35·19-s + 2.24·21-s + i·23-s + (−2.62 − 4.25i)25-s − 2.84i·27-s + 3.66·29-s − 4.46·31-s + ⋯
L(s)  = 1  − 0.284i·3-s + (0.487 − 0.872i)5-s + 1.71i·7-s + 0.918·9-s − 1.39·11-s + 1.53i·13-s + (−0.248 − 0.138i)15-s + 1.73i·17-s − 0.311·19-s + 0.489·21-s + 0.208i·23-s + (−0.524 − 0.851i)25-s − 0.546i·27-s + 0.680·29-s − 0.802·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.487 - 0.872i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29541 + 0.760103i\)
\(L(\frac12)\) \(\approx\) \(1.29541 + 0.760103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.09 + 1.95i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.493iT - 3T^{2} \)
7 \( 1 - 4.54iT - 7T^{2} \)
11 \( 1 + 4.61T + 11T^{2} \)
13 \( 1 - 5.54iT - 13T^{2} \)
17 \( 1 - 7.16iT - 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
29 \( 1 - 3.66T + 29T^{2} \)
31 \( 1 + 4.46T + 31T^{2} \)
37 \( 1 - 3.32iT - 37T^{2} \)
41 \( 1 - 8.95T + 41T^{2} \)
43 \( 1 + 8.68iT - 43T^{2} \)
47 \( 1 - 9.59iT - 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 - 2.08T + 59T^{2} \)
61 \( 1 + 0.686T + 61T^{2} \)
67 \( 1 - 8.84iT - 67T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 + 4.44iT - 73T^{2} \)
79 \( 1 - 8.65T + 79T^{2} \)
83 \( 1 - 4.43iT - 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 1.21iT - 97T^{2} \)
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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

−10.04814631579991297263473556858, −9.301080898392869686264131442933, −8.575771052910472818813160301182, −7.948789154682311407663619458053, −6.62940368259969221662859671923, −5.85221230260125400215121578182, −5.07353419126701956768884082428, −4.10306584882352276051268379523, −2.34226709096507556304382769051, −1.73062528074813721344556672454, 0.70936609504300154655545125035, 2.56534371717042132370411480631, 3.47767761305605488956825737111, 4.62268327642009038040515053250, 5.46549487327842845217509541198, 6.73837940707105019560340949081, 7.47485310486039748682843172731, 7.84160779425613226169548746671, 9.516927802345416724536338661343, 10.09314233227010804287681342623

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