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Properties

Label 2-1664-16.13-c1-0-32
Degree $2$
Conductor $1664$
Sign $-0.394 + 0.918i$
Analytic cond. $13.2871$
Root an. cond. $3.64514$
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  + (0.718 + 0.718i)3-s + (−2.02 + 2.02i)5-s − 0.407i·7-s − 1.96i·9-s + (−1.76 + 1.76i)11-s + (0.707 + 0.707i)13-s − 2.90·15-s − 4.09·17-s + (−2.13 − 2.13i)19-s + (0.292 − 0.292i)21-s − 2.42i·23-s − 3.16i·25-s + (3.56 − 3.56i)27-s + (−2.51 − 2.51i)29-s − 0.736·31-s + ⋯
L(s)  = 1  + (0.414 + 0.414i)3-s + (−0.903 + 0.903i)5-s − 0.154i·7-s − 0.656i·9-s + (−0.530 + 0.530i)11-s + (0.196 + 0.196i)13-s − 0.749·15-s − 0.991·17-s + (−0.490 − 0.490i)19-s + (0.0639 − 0.0639i)21-s − 0.506i·23-s − 0.632i·25-s + (0.686 − 0.686i)27-s + (−0.466 − 0.466i)29-s − 0.132·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(13.2871\)
Root analytic conductor: \(3.64514\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3577172734\)
\(L(\frac12)\) \(\approx\) \(0.3577172734\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.718 - 0.718i)T + 3iT^{2} \)
5 \( 1 + (2.02 - 2.02i)T - 5iT^{2} \)
7 \( 1 + 0.407iT - 7T^{2} \)
11 \( 1 + (1.76 - 1.76i)T - 11iT^{2} \)
17 \( 1 + 4.09T + 17T^{2} \)
19 \( 1 + (2.13 + 2.13i)T + 19iT^{2} \)
23 \( 1 + 2.42iT - 23T^{2} \)
29 \( 1 + (2.51 + 2.51i)T + 29iT^{2} \)
31 \( 1 + 0.736T + 31T^{2} \)
37 \( 1 + (1.13 - 1.13i)T - 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (0.341 - 0.341i)T - 43iT^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + (-6.62 + 6.62i)T - 53iT^{2} \)
59 \( 1 + (9.01 - 9.01i)T - 59iT^{2} \)
61 \( 1 + (-2.36 - 2.36i)T + 61iT^{2} \)
67 \( 1 + (-0.526 - 0.526i)T + 67iT^{2} \)
71 \( 1 + 7.15iT - 71T^{2} \)
73 \( 1 - 5.01iT - 73T^{2} \)
79 \( 1 + 9.29T + 79T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−8.976630866579169010036001393840, −8.471743060072737756485580354840, −7.35758182055396340428379384158, −6.94104231364857840984500425172, −6.02071670822251527163438589140, −4.64667578185427130425257149756, −4.00070016428703078883299836695, −3.17793935350672034628078667122, −2.21449848799867458913006877769, −0.12701529489254712028253784245, 1.43456655336767613606091858407, 2.61984544550887397455079718046, 3.71913580084186100520042079179, 4.66804615469831048612617066192, 5.39223914681166865971707096522, 6.48011877907432568852882275759, 7.51994552323341914137892309765, 8.140577412013834832561946967951, 8.547161739826803587227119499440, 9.353885662595483627993979268013

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