Properties

Label 2-920-5.4-c1-0-25
Degree $2$
Conductor $920$
Sign $-0.168 + 0.985i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.356i·3-s + (0.376 − 2.20i)5-s − 2.46i·7-s + 2.87·9-s + 1.61·11-s − 2.62i·13-s + (−0.785 − 0.134i)15-s + 2.58i·17-s − 4.02·19-s − 0.878·21-s + i·23-s + (−4.71 − 1.65i)25-s − 2.09i·27-s + 7.08·29-s − 4.58·31-s + ⋯
L(s)  = 1  − 0.205i·3-s + (0.168 − 0.985i)5-s − 0.931i·7-s + 0.957·9-s + 0.487·11-s − 0.728i·13-s + (−0.202 − 0.0346i)15-s + 0.627i·17-s − 0.923·19-s − 0.191·21-s + 0.208i·23-s + (−0.943 − 0.331i)25-s − 0.402i·27-s + 1.31·29-s − 0.823·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.168 + 0.985i$
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: \((2,\ 920,\ (\ :1/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05380 - 1.24893i\)
\(L(\frac12)\) \(\approx\) \(1.05380 - 1.24893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.376 + 2.20i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.356iT - 3T^{2} \)
7 \( 1 + 2.46iT - 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 - 2.58iT - 17T^{2} \)
19 \( 1 + 4.02T + 19T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 2.96iT - 37T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 + 2.30iT - 43T^{2} \)
47 \( 1 + 6.88iT - 47T^{2} \)
53 \( 1 + 6.76iT - 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 - 9.25T + 61T^{2} \)
67 \( 1 - 15.7iT - 67T^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 - 6.03iT - 73T^{2} \)
79 \( 1 - 1.35T + 79T^{2} \)
83 \( 1 + 8.59iT - 83T^{2} \)
89 \( 1 - 8.84T + 89T^{2} \)
97 \( 1 + 8.01iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12985566865501574659036750871, −8.896535236740795019298026212540, −8.239099585407007754368091423777, −7.29715948927124687803687852337, −6.51733356743388529764650207641, −5.41333861686902724138172428447, −4.39048881424680461662240242204, −3.73366149073663417467170563976, −1.89759808271891757399838504772, −0.813099031822065403653861895907, 1.80494337868896314017923851487, 2.85227862607201372036425427034, 4.02914924577921089580235413062, 4.97913390017172419708668479224, 6.27123942616005609087928573102, 6.71873195169460356613924500547, 7.69635587893771773277357255576, 8.841228739558967446460604627488, 9.490850213373620302120742384158, 10.29599892239293236990260091064

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