Properties

Label 2-896-16.13-c1-0-14
Degree $2$
Conductor $896$
Sign $0.816 + 0.577i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (0.599 + 0.599i)3-s + (−0.974 + 0.974i)5-s i·7-s − 2.28i·9-s + (1.72 − 1.72i)11-s + (−1.90 − 1.90i)13-s − 1.16·15-s + 6.71·17-s + (−2.94 − 2.94i)19-s + (0.599 − 0.599i)21-s − 5.29i·23-s + 3.09i·25-s + (3.16 − 3.16i)27-s + (3.03 + 3.03i)29-s + 1.19·31-s + ⋯
L(s)  = 1  + (0.346 + 0.346i)3-s + (−0.436 + 0.436i)5-s − 0.377i·7-s − 0.760i·9-s + (0.519 − 0.519i)11-s + (−0.528 − 0.528i)13-s − 0.302·15-s + 1.62·17-s + (−0.676 − 0.676i)19-s + (0.130 − 0.130i)21-s − 1.10i·23-s + 0.619i·25-s + (0.609 − 0.609i)27-s + (0.562 + 0.562i)29-s + 0.215·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52138 - 0.483579i\)
\(L(\frac12)\) \(\approx\) \(1.52138 - 0.483579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + (-0.599 - 0.599i)T + 3iT^{2} \)
5 \( 1 + (0.974 - 0.974i)T - 5iT^{2} \)
11 \( 1 + (-1.72 + 1.72i)T - 11iT^{2} \)
13 \( 1 + (1.90 + 1.90i)T + 13iT^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 + (2.94 + 2.94i)T + 19iT^{2} \)
23 \( 1 + 5.29iT - 23T^{2} \)
29 \( 1 + (-3.03 - 3.03i)T + 29iT^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 + (-7.02 + 7.02i)T - 43iT^{2} \)
47 \( 1 + 3.06T + 47T^{2} \)
53 \( 1 + (-3.01 + 3.01i)T - 53iT^{2} \)
59 \( 1 + (4.96 - 4.96i)T - 59iT^{2} \)
61 \( 1 + (-9.69 - 9.69i)T + 61iT^{2} \)
67 \( 1 + (3.55 + 3.55i)T + 67iT^{2} \)
71 \( 1 + 11.5iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 4.06T + 79T^{2} \)
83 \( 1 + (9.17 + 9.17i)T + 83iT^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 + 2.51T + 97T^{2} \)
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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

−10.11694819999019411617248331402, −9.177601729517991822684703740595, −8.426858763186084115798030400732, −7.48212466426158811077041254112, −6.70151636009945957150588615718, −5.72281458863592554082137619182, −4.47946067279548672184894860436, −3.55584232632482166547017918155, −2.82280997273463673014195018821, −0.818174801406378899699886669253, 1.43842225129493109868235269547, 2.58412696539106938957295299688, 3.91425067544564972436480251410, 4.84240847812902084339112012432, 5.81891016779094809501155094880, 6.93823509282059817238728594438, 7.899752592536011456012591073009, 8.234295801127825386868744269525, 9.424225553744331797629358897877, 9.994027139038798213163014889142

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