Properties

Label 2-896-8.5-c1-0-21
Degree $2$
Conductor $896$
Sign $-1$
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  − 2.61i·3-s + 2.61i·5-s − 7-s − 3.82·9-s − 2.16i·11-s − 0.448i·13-s + 6.82·15-s − 7.65·17-s − 4.77i·19-s + 2.61i·21-s − 6.82·23-s − 1.82·25-s + 2.16i·27-s − 9.55i·29-s − 5.65·31-s + ⋯
L(s)  = 1  − 1.50i·3-s + 1.16i·5-s − 0.377·7-s − 1.27·9-s − 0.652i·11-s − 0.124i·13-s + 1.76·15-s − 1.85·17-s − 1.09i·19-s + 0.570i·21-s − 1.42·23-s − 0.365·25-s + 0.416i·27-s − 1.77i·29-s − 1.01·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-1$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612968i\)
\(L(\frac12)\) \(\approx\) \(0.612968i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 + 2.61iT - 3T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
11 \( 1 + 2.16iT - 11T^{2} \)
13 \( 1 + 0.448iT - 13T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 + 4.77iT - 19T^{2} \)
23 \( 1 + 6.82T + 23T^{2} \)
29 \( 1 + 9.55iT - 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 - 5.22iT - 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 - 2.16iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 - 0.448iT - 59T^{2} \)
61 \( 1 - 12.1iT - 61T^{2} \)
67 \( 1 + 3.06iT - 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.702436422274549485008242724931, −8.618045215428539080279714614226, −7.85973483256451630743230887300, −6.85433464557324392681858582816, −6.60956141677591059337507567762, −5.75409691354189271580749715436, −4.10580429523943148614150629175, −2.77590401298189243668772969266, −2.11700464203147875685397709880, −0.26990162487436964217273353815, 1.96274941220331534526711784378, 3.64335159958302093102705607241, 4.31907129274069995006206391116, 5.02969442222668144545760792905, 5.91178159885015951601624762899, 7.15124248992741200794844241104, 8.432245271454007105459325237274, 9.061872179933319466927883533908, 9.556512434357397986521729459504, 10.46738928365621175626931574035

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