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Properties

Label 2-896-112.13-c0-0-1
Degree $2$
Conductor $896$
Sign $0.923 + 0.382i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
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·7-s + i·9-s + (1 − i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (−1 + i)43-s − 49-s + (−1 + i)53-s + 63-s + (−1 − i)67-s + 2i·71-s + (−1 − i)77-s − 81-s + (1 + i)99-s + ⋯
L(s)  = 1  − i·7-s + i·9-s + (1 − i)11-s − i·25-s + (1 + i)29-s + (1 − i)37-s + (−1 + i)43-s − 49-s + (−1 + i)53-s + 63-s + (−1 − i)67-s + 2i·71-s + (−1 − i)77-s − 81-s + (1 + i)99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.042883170\)
\(L(\frac12)\) \(\approx\) \(1.042883170\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
5 \( 1 + iT^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.43096756719523208032377652804, −9.466837397912958160309878311214, −8.482925147518149956285616867902, −7.81229612672820194244209303087, −6.83070471734107728099826645424, −6.05282413147094061700808480869, −4.83454051071575640101649654839, −4.01707793218408609635329984241, −2.87648605440487255235932315351, −1.28183308675642035097784966116, 1.62432983907612950363498786813, 2.96291420653715139077888087295, 4.07256301127090740614578750652, 5.08929932523714480970487642022, 6.23265098550158758680943743417, 6.74456999986489418142346205143, 7.906721637694024845984422229152, 8.898482236773261366700062761677, 9.457780361990515816240673801009, 10.12646156634763419906207487499

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