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Properties

Label 2-160-20.7-c5-0-26
Degree $2$
Conductor $160$
Sign $-0.885 + 0.464i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.56 − 2.56i)3-s + (3.91 − 55.7i)5-s + (86.5 − 86.5i)7-s − 229. i·9-s − 138. i·11-s + (−393. + 393. i)13-s + (−153. + 133. i)15-s + (724. + 724. i)17-s + 1.46e3·19-s − 444.·21-s + (−2.85e3 − 2.85e3i)23-s + (−3.09e3 − 436. i)25-s + (−1.21e3 + 1.21e3i)27-s + 2.51e3i·29-s − 2.96e3i·31-s + ⋯
L(s)  = 1  + (−0.164 − 0.164i)3-s + (0.0700 − 0.997i)5-s + (0.667 − 0.667i)7-s − 0.945i·9-s − 0.344i·11-s + (−0.645 + 0.645i)13-s + (−0.175 + 0.152i)15-s + (0.607 + 0.607i)17-s + 0.930·19-s − 0.220·21-s + (−1.12 − 1.12i)23-s + (−0.990 − 0.139i)25-s + (−0.320 + 0.320i)27-s + 0.555i·29-s − 0.554i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.885 + 0.464i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.885 + 0.464i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.370318258\)
\(L(\frac12)\) \(\approx\) \(1.370318258\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.91 + 55.7i)T \)
good3 \( 1 + (2.56 + 2.56i)T + 243iT^{2} \)
7 \( 1 + (-86.5 + 86.5i)T - 1.68e4iT^{2} \)
11 \( 1 + 138. iT - 1.61e5T^{2} \)
13 \( 1 + (393. - 393. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-724. - 724. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.46e3T + 2.47e6T^{2} \)
23 \( 1 + (2.85e3 + 2.85e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 2.51e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.96e3iT - 2.86e7T^{2} \)
37 \( 1 + (1.45e3 + 1.45e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + (4.66e3 + 4.66e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-2.07e4 + 2.07e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.15e4 - 1.15e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 3.26e4T + 7.14e8T^{2} \)
61 \( 1 + 1.11e4T + 8.44e8T^{2} \)
67 \( 1 + (1.53e4 - 1.53e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 7.24e4iT - 1.80e9T^{2} \)
73 \( 1 + (6.69e3 - 6.69e3i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.66e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.57e4 - 6.57e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 5.58e4iT - 5.58e9T^{2} \)
97 \( 1 + (-5.01e4 - 5.01e4i)T + 8.58e9iT^{2} \)
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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

−11.95046739437152782283517462993, −10.51560671364558944580165107127, −9.464998810279881079635293819151, −8.456966961927042961726753773677, −7.41065215336081508619196474858, −6.09442957517956192276926409519, −4.87075567730748632818579412078, −3.75655139210449209207278569361, −1.63460240928748611426302421262, −0.45682829701277687483776477033, 1.92116645520560213442555609935, 3.13437255440067169987592080004, 4.92067533599916355569011486231, 5.78325632208015428824836835390, 7.37411177394028334206244005608, 7.991801532383061857402899116966, 9.637013350230052253851945740713, 10.36361617430524381740779728318, 11.46489722289101973694772766571, 12.09923309053950767034113082857

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