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Properties

Label 2-320-40.3-c1-0-6
Degree $2$
Conductor $320$
Sign $0.522 + 0.852i$
Analytic cond. $2.55521$
Root an. cond. $1.59850$
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  + (−0.456 − 0.456i)3-s + (2.18 + 0.456i)5-s + (−2.79 − 2.79i)7-s − 2.58i·9-s + 4.37·11-s + (−1.73 + 1.73i)13-s + (−0.791 − 1.20i)15-s + (3 − 3i)17-s − 3.46i·19-s + 2.55i·21-s + (−0.791 + 0.791i)23-s + (4.58 + 1.99i)25-s + (−2.55 + 2.55i)27-s + 5.29·29-s − 1.58i·31-s + ⋯
L(s)  = 1  + (−0.263 − 0.263i)3-s + (0.978 + 0.204i)5-s + (−1.05 − 1.05i)7-s − 0.860i·9-s + 1.31·11-s + (−0.480 + 0.480i)13-s + (−0.204 − 0.312i)15-s + (0.727 − 0.727i)17-s − 0.794i·19-s + 0.556i·21-s + (−0.164 + 0.164i)23-s + (0.916 + 0.399i)25-s + (−0.490 + 0.490i)27-s + 0.982·29-s − 0.284i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13567 - 0.636091i\)
\(L(\frac12)\) \(\approx\) \(1.13567 - 0.636091i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.456i)T \)
good3 \( 1 + (0.456 + 0.456i)T + 3iT^{2} \)
7 \( 1 + (2.79 + 2.79i)T + 7iT^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (0.791 - 0.791i)T - 23iT^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 1.58iT - 31T^{2} \)
37 \( 1 + (5.19 + 5.19i)T + 37iT^{2} \)
41 \( 1 + 7.58T + 41T^{2} \)
43 \( 1 + (-8.29 - 8.29i)T + 43iT^{2} \)
47 \( 1 + (-0.791 - 0.791i)T + 47iT^{2} \)
53 \( 1 + (2.64 - 2.64i)T - 53iT^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 - 9.66iT - 61T^{2} \)
67 \( 1 + (8.29 - 8.29i)T - 67iT^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (-0.582 - 0.582i)T + 73iT^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (4.83 + 4.83i)T + 83iT^{2} \)
89 \( 1 - 3.16iT - 89T^{2} \)
97 \( 1 + (-0.582 + 0.582i)T - 97iT^{2} \)
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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

−11.60537416940403308133765004850, −10.37526967242800635728219757041, −9.576355437309010828651576291404, −9.090071859657474306841660042453, −7.10974404488322538742096957428, −6.76234129169854604559939096921, −5.82203979151618119990072089480, −4.23458053509013375024023428511, −3.02196512345690803106487342514, −1.07028622373396869751093103667, 1.92554315144311601114371901541, 3.35546001902201512383705057784, 4.99797413585837926245795086769, 5.89505450297657342136389236909, 6.59554878058396533160034760266, 8.198625402990153660214975597408, 9.152423506336126777094516974725, 9.948551458618729811107302367865, 10.57002502485180181265324483685, 12.17245040808658280089626484929

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