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Properties

Label 1-320-320.59-r1-0-0
Degree $1$
Conductor $320$
Sign $0.956 - 0.290i$
Analytic cond. $34.3887$
Root an. cond. $34.3887$
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  + (0.382 + 0.923i)3-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.923 − 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (0.923 − 0.382i)19-s + (0.382 − 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.923 − 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (0.923 + 0.382i)37-s + (−0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(34.3887\)
Root analytic conductor: \(34.3887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (1:\ ),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.736589415 - 0.2575987058i\)
\(L(\frac12)\) \(\approx\) \(1.736589415 - 0.2575987058i\)
\(L(1)\) \(\approx\) \(1.112527256 + 0.1207026223i\)
\(L(1)\) \(\approx\) \(1.112527256 + 0.1207026223i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 + 0.382i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.97442957563460676298148582434, −24.38100411316169948993667346604, −22.94757725959233577390138682536, −22.658126416407847995254932127062, −21.38364274858085523046225464305, −20.12427397558731812519582283195, −19.70068422326137653575182003849, −18.62837442070083445869722835978, −17.96804690445000551464335364144, −17.00624553922243243922105770573, −15.77047182781308115743695406039, −14.84959406848930393676411226903, −14.00341621765790055369397351367, −12.83623712624076596937091847329, −12.31602494514123605749636752200, −11.41048789397275397438640230398, −9.6236282312003440365853492426, −9.2931424474437033849109220830, −7.79234584868975961528932548771, −7.12639473608895015213801682289, −6.06256735685781753174390134140, −4.918163396804478369495883627670, −3.200241382824792066701837184101, −2.437635649162033805494059817640, −1.03439991904390446979827706314, 0.58340506271557160361289855568, 2.54258554526588633932854334037, 3.588048695391082867732005929473, 4.43933509870482644100628353757, 5.703165451442741170415929383525, 6.877245319652340579751361660039, 8.096889777661713698774738514583, 9.14946804327726502973662611234, 9.945502978681985298310763358622, 10.79739381775252248889644895788, 11.823659655049811793452369518351, 13.19937155964340912020674911151, 13.98851864204609577738018362532, 14.87250000614672430966333263231, 15.86285309838594219903297664960, 16.74759823888892112504099428861, 17.236222512951637651225559707752, 18.99858287514244184762610547904, 19.52802584376040328881665488744, 20.398868229717673278889745923227, 21.3823589418495684475414182817, 22.15764807493837478062683043803, 22.8410579675185199225964265902, 24.087720099531787887296178108909, 24.90195477583150925202701586035

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