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Properties

Label 2-1664-104.5-c0-0-1
Degree $2$
Conductor $1664$
Sign $-0.289 - 0.957i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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·3-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s + (−1 + i)11-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + i·17-s + (−0.707 + 0.707i)21-s − 1.41i·23-s + i·27-s − 1.41i·29-s + (−1 − i)33-s + 1.00i·35-s + (0.707 − 0.707i)37-s + (0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + i·3-s + (0.707 + 0.707i)5-s + (0.707 + 0.707i)7-s + (−1 + i)11-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + i·17-s + (−0.707 + 0.707i)21-s − 1.41i·23-s + i·27-s − 1.41i·29-s + (−1 − i)33-s + 1.00i·35-s + (0.707 − 0.707i)37-s + (0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288733841\)
\(L(\frac12)\) \(\approx\) \(1.288733841\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + iT^{2} \)
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

−9.919444593375320371812957275797, −9.247626409070677014792444475769, −8.178089996160664676521972801323, −7.56655516283827009206076450563, −6.42934681294388666830456057040, −5.58879484636940979753433714076, −4.83984312061834445814608165338, −4.12546933301071398555441832496, −2.66344206676534705024145389274, −2.14855686489585693498138425821, 1.06619462788161135981329501116, 1.90950780038349182791951989819, 3.11634578268721646686176094240, 4.59491423057164344259054663053, 5.21047905721699790842717326679, 6.08147011726916074779471198215, 7.21000614380711180730592440568, 7.54426032960100410676385187418, 8.410502589285058413929771320963, 9.306394080164984834495808645854

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