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Properties

Label 2-320-40.27-c1-0-11
Degree $2$
Conductor $320$
Sign $0.0859 + 0.996i$
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  + (2.18 − 2.18i)3-s + (0.456 − 2.18i)5-s + (−1.79 + 1.79i)7-s − 6.58i·9-s − 0.913·11-s + (1.73 + 1.73i)13-s + (−3.79 − 5.79i)15-s + (3 + 3i)17-s + 3.46i·19-s + 7.84i·21-s + (−3.79 − 3.79i)23-s + (−4.58 − 1.99i)25-s + (−7.84 − 7.84i)27-s + 5.29·29-s + 7.58i·31-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)3-s + (0.204 − 0.978i)5-s + (−0.677 + 0.677i)7-s − 2.19i·9-s − 0.275·11-s + (0.480 + 0.480i)13-s + (−0.978 − 1.49i)15-s + (0.727 + 0.727i)17-s + 0.794i·19-s + 1.71i·21-s + (−0.790 − 0.790i)23-s + (−0.916 − 0.399i)25-s + (−1.50 − 1.50i)27-s + 0.982·29-s + 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0859 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0859 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.0859 + 0.996i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.0859 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37005 - 1.25700i\)
\(L(\frac12)\) \(\approx\) \(1.37005 - 1.25700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.456 + 2.18i)T \)
good3 \( 1 + (-2.18 + 2.18i)T - 3iT^{2} \)
7 \( 1 + (1.79 - 1.79i)T - 7iT^{2} \)
11 \( 1 + 0.913T + 11T^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + (3.79 + 3.79i)T + 23iT^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 7.58iT - 31T^{2} \)
37 \( 1 + (-5.19 + 5.19i)T - 37iT^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 + (-0.361 + 0.361i)T - 43iT^{2} \)
47 \( 1 + (-3.79 + 3.79i)T - 47iT^{2} \)
53 \( 1 + (2.64 + 2.64i)T + 53iT^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 6.20iT - 61T^{2} \)
67 \( 1 + (0.361 + 0.361i)T + 67iT^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 + (8.58 - 8.58i)T - 73iT^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \)
89 \( 1 - 15.1iT - 89T^{2} \)
97 \( 1 + (8.58 + 8.58i)T + 97iT^{2} \)
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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

−12.03845318839149946865211819440, −10.18723979610416810874609842327, −9.177479028460119271347700176665, −8.506348948298229196120340957737, −7.896956382810573937051708029147, −6.58719538313365916483394971128, −5.75410502229368209560226775472, −3.92907818449291946471738704722, −2.61843304996480414060947674664, −1.40444699084317220501199822112, 2.70333672984345784861745328648, 3.37701364160062147997722977312, 4.42921306707228677405983253199, 5.91296672433165538751030943548, 7.31572903121117381276883363488, 8.087121133299872806761809286171, 9.374084328060229330894989894436, 9.941136514923740115296648615802, 10.54088829445099976289217024920, 11.50818316472228896773475801142

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