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Properties

Label 2-320-5.2-c0-0-0
Degree $2$
Conductor $320$
Sign $0.850 - 0.525i$
Analytic cond. $0.159700$
Root an. cond. $0.399625$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + i·9-s + (1 − i)13-s + (−1 − i)17-s − 25-s + (−1 − i)37-s − 45-s − i·49-s + (−1 + i)53-s + (1 + i)65-s + (1 − i)73-s − 81-s + (1 − i)85-s + (1 + i)97-s + 2·101-s + ⋯
L(s)  = 1  + i·5-s + i·9-s + (1 − i)13-s + (−1 − i)17-s − 25-s + (−1 − i)37-s − 45-s − i·49-s + (−1 + i)53-s + (1 + i)65-s + (1 − i)73-s − 81-s + (1 − i)85-s + (1 + i)97-s + 2·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.159700\)
Root analytic conductor: \(0.399625\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7909838631\)
\(L(\frac12)\) \(\approx\) \(0.7909838631\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−11.70730757930274210052070860578, −10.85084804999473395270961997714, −10.42385273045418601749797180161, −9.137370007529070708115527314309, −8.032732310269770728252854160536, −7.18830048997100790055883055519, −6.12915681482321798350687279061, −4.99971014154535995678329749748, −3.52007095924675564889294389541, −2.31263235827859086969458246725, 1.59942077213077209988873457183, 3.67342203931195572136873608633, 4.58466241709766521529310591437, 5.98658972706148741961646032438, 6.76267191062233945982580644459, 8.315241678892954349982762143012, 8.887964672811984972068635532586, 9.718829672758755078742899264270, 10.99823526752159599260567792653, 11.83342670478939600778000367090

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