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Properties

Label 2-320-320.27-c1-0-45
Degree $2$
Conductor $320$
Sign $-0.463 + 0.886i$
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  + (0.660 − 1.25i)2-s + (0.924 − 0.183i)3-s + (−1.12 − 1.65i)4-s + (1.84 − 1.26i)5-s + (0.380 − 1.27i)6-s + (−1.57 − 3.79i)7-s + (−2.81 + 0.321i)8-s + (−1.95 + 0.808i)9-s + (−0.369 − 3.14i)10-s + (3.51 + 5.25i)11-s + (−1.34 − 1.31i)12-s + (−0.166 − 0.110i)13-s + (−5.78 − 0.539i)14-s + (1.46 − 1.51i)15-s + (−1.45 + 3.72i)16-s − 4.50·17-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)2-s + (0.533 − 0.106i)3-s + (−0.564 − 0.825i)4-s + (0.823 − 0.566i)5-s + (0.155 − 0.521i)6-s + (−0.593 − 1.43i)7-s + (−0.993 + 0.113i)8-s + (−0.650 + 0.269i)9-s + (−0.116 − 0.993i)10-s + (1.05 + 1.58i)11-s + (−0.388 − 0.380i)12-s + (−0.0460 − 0.0307i)13-s + (−1.54 − 0.144i)14-s + (0.379 − 0.390i)15-s + (−0.363 + 0.931i)16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $-0.463 + 0.886i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ -0.463 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979764 - 1.61730i\)
\(L(\frac12)\) \(\approx\) \(0.979764 - 1.61730i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.660 + 1.25i)T \)
5 \( 1 + (-1.84 + 1.26i)T \)
good3 \( 1 + (-0.924 + 0.183i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (1.57 + 3.79i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-3.51 - 5.25i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (0.166 + 0.110i)T + (4.97 + 12.0i)T^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
19 \( 1 + (-7.57 + 1.50i)T + (17.5 - 7.27i)T^{2} \)
23 \( 1 + (-2.46 - 1.02i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-2.21 + 3.31i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 - 1.97T + 31T^{2} \)
37 \( 1 + (-0.230 - 0.344i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.84 + 0.766i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (0.204 - 1.03i)T + (-39.7 - 16.4i)T^{2} \)
47 \( 1 - 4.20iT - 47T^{2} \)
53 \( 1 + (-0.0594 + 0.299i)T + (-48.9 - 20.2i)T^{2} \)
59 \( 1 + (2.06 - 10.3i)T + (-54.5 - 22.5i)T^{2} \)
61 \( 1 + (4.20 - 6.29i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.50 + 7.58i)T + (-61.8 + 25.6i)T^{2} \)
71 \( 1 + (-9.41 - 3.89i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (9.42 + 3.90i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (2.74 + 2.74i)T + 79iT^{2} \)
83 \( 1 + (-2.74 - 1.83i)T + (31.7 + 76.6i)T^{2} \)
89 \( 1 + (2.80 - 6.77i)T + (-62.9 - 62.9i)T^{2} \)
97 \( 1 + (0.499 - 0.499i)T - 97iT^{2} \)
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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

−11.40221549513493631781896273933, −10.28230836851734205700608881184, −9.547296704196415407662688600351, −9.026412721495727411502137902926, −7.40496805235999791756897752799, −6.36623470526805036552469344653, −4.94630619262655821166676251635, −4.07134602298506042671705467593, −2.69004173036098012295643425427, −1.29580146519704691083680123827, 2.80438992746901401622754718234, 3.41347866540959342480637939332, 5.39222925164931913308592051308, 6.06354870511369782368378024998, 6.75272106235766705303806699845, 8.364767945636144062394943866289, 9.046563837657800792307110543381, 9.492793642052522810569307417685, 11.28701475876202030969417125456, 11.99161162688644595392068306314

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