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Properties

Label 2-920-5.4-c1-0-6
Degree $2$
Conductor $920$
Sign $-0.653 - 0.757i$
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  + 1.78i·3-s + (1.46 + 1.69i)5-s − 1.75i·7-s − 0.192·9-s − 4.77·11-s + 1.72i·13-s + (−3.02 + 2.61i)15-s + 7.81i·17-s + 2.43·19-s + 3.13·21-s − i·23-s + (−0.731 + 4.94i)25-s + 5.01i·27-s − 7.86·29-s + 6.14·31-s + ⋯
L(s)  = 1  + 1.03i·3-s + (0.653 + 0.757i)5-s − 0.663i·7-s − 0.0640·9-s − 1.43·11-s + 0.479i·13-s + (−0.780 + 0.673i)15-s + 1.89i·17-s + 0.559·19-s + 0.684·21-s − 0.208i·23-s + (−0.146 + 0.989i)25-s + 0.965i·27-s − 1.46·29-s + 1.10·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.653 - 0.757i$
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: \((2,\ 920,\ (\ :1/2),\ -0.653 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611954 + 1.33641i\)
\(L(\frac12)\) \(\approx\) \(0.611954 + 1.33641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.46 - 1.69i)T \)
23 \( 1 + iT \)
good3 \( 1 - 1.78iT - 3T^{2} \)
7 \( 1 + 1.75iT - 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 - 1.72iT - 13T^{2} \)
17 \( 1 - 7.81iT - 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 6.14T + 31T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 + 3.26iT - 43T^{2} \)
47 \( 1 + 8.46iT - 47T^{2} \)
53 \( 1 + 2.76iT - 53T^{2} \)
59 \( 1 + 1.91T + 59T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 0.0111iT - 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 + 2.64iT - 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−10.35212987194975860216403107416, −9.940417976822903033233289266845, −8.871964705089737847531596489496, −7.84243063616016471915740606826, −6.98535504871000996914388644397, −5.96465424770682619387971360247, −5.09365749652270624749351483968, −4.06138947718248227756667366633, −3.21569636895417067068233725685, −1.87356140930891123746607580634, 0.67982568968155642874172807630, 2.08413100349916062322362818818, 2.89021446239375467108641150774, 4.79554706358359470418346243442, 5.43248961304904939542159355541, 6.23869652047303499372452869136, 7.54347776741841818274737505809, 7.78026748421824141143834624659, 9.018786265847875297283125518843, 9.595190550696568845421774629930

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