Properties

Label 2-320-320.267-c1-0-11
Degree $2$
Conductor $320$
Sign $0.894 - 0.446i$
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  + (−1.26 − 0.622i)2-s + (0.562 + 0.376i)3-s + (1.22 + 1.58i)4-s + (1.23 + 1.86i)5-s + (−0.480 − 0.828i)6-s + (0.396 − 0.164i)7-s + (−0.568 − 2.77i)8-s + (−0.972 − 2.34i)9-s + (−0.405 − 3.13i)10-s + (3.28 + 0.653i)11-s + (0.0938 + 1.35i)12-s + (0.950 + 4.77i)13-s + (−0.604 − 0.0384i)14-s + (−0.00643 + 1.51i)15-s + (−1.00 + 3.87i)16-s − 3.35·17-s + ⋯
L(s)  = 1  + (−0.897 − 0.440i)2-s + (0.324 + 0.217i)3-s + (0.611 + 0.790i)4-s + (0.552 + 0.833i)5-s + (−0.196 − 0.338i)6-s + (0.149 − 0.0620i)7-s + (−0.200 − 0.979i)8-s + (−0.324 − 0.782i)9-s + (−0.128 − 0.991i)10-s + (0.990 + 0.197i)11-s + (0.0271 + 0.389i)12-s + (0.263 + 1.32i)13-s + (−0.161 − 0.0102i)14-s + (−0.00166 + 0.390i)15-s + (−0.251 + 0.967i)16-s − 0.812·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.894 - 0.446i$
Analytic conductor: \(2.55521\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :1/2),\ 0.894 - 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06768 + 0.251683i\)
\(L(\frac12)\) \(\approx\) \(1.06768 + 0.251683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 + 0.622i)T \)
5 \( 1 + (-1.23 - 1.86i)T \)
good3 \( 1 + (-0.562 - 0.376i)T + (1.14 + 2.77i)T^{2} \)
7 \( 1 + (-0.396 + 0.164i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-3.28 - 0.653i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + (-0.950 - 4.77i)T + (-12.0 + 4.97i)T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + (-2.30 - 1.54i)T + (7.27 + 17.5i)T^{2} \)
23 \( 1 + (0.570 - 1.37i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-3.65 + 0.727i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 - 9.38T + 31T^{2} \)
37 \( 1 + (4.14 + 0.823i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.549 + 1.32i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.00 - 1.50i)T + (-16.4 + 39.7i)T^{2} \)
47 \( 1 + 0.982iT - 47T^{2} \)
53 \( 1 + (3.34 + 5.00i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (4.02 + 6.02i)T + (-22.5 + 54.5i)T^{2} \)
61 \( 1 + (0.761 - 0.151i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (2.60 - 3.89i)T + (-25.6 - 61.8i)T^{2} \)
71 \( 1 + (-5.09 + 12.3i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-5.78 + 13.9i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-7.34 - 7.34i)T + 79iT^{2} \)
83 \( 1 + (1.49 + 7.52i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (14.2 + 5.92i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (7.61 - 7.61i)T - 97iT^{2} \)
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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

−11.60836973284876453922389093551, −10.69758061245820238378803678169, −9.568439385456015296525292346710, −9.264909251294231544067494237862, −8.157139696030060683573943576804, −6.73188342487115780810266253409, −6.41153618679087661476299358645, −4.17136706730358100142890481895, −3.08040338547015219572860802075, −1.71387476451845938549599261581, 1.17247908330470269196660323148, 2.63349468810626748568523441247, 4.81387485642035757502558335415, 5.78596679336257025426124543666, 6.81934844387132506836029273958, 8.192530409304294301599892126563, 8.499916720684379648134607480357, 9.515460190316387803824030011057, 10.42974748549880635129578679454, 11.37101339999065018930718829612

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