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Properties

Label 2-896-56.3-c1-0-3
Degree $2$
Conductor $896$
Sign $-0.773 - 0.633i$
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  + (−0.753 − 0.435i)3-s + (0.866 + 1.5i)5-s + (0.615 + 2.57i)7-s + (−1.12 − 1.94i)9-s + (−1.81 + 3.15i)11-s − 1.01·13-s − 1.50i·15-s + (−0.621 − 0.358i)17-s + (−5.45 + 3.15i)19-s + (0.655 − 2.20i)21-s + (−3.15 + 1.81i)23-s + (1 − 1.73i)25-s + 4.56i·27-s − 4.58i·29-s + (3.76 − 6.52i)31-s + ⋯
L(s)  = 1  + (−0.435 − 0.251i)3-s + (0.387 + 0.670i)5-s + (0.232 + 0.972i)7-s + (−0.373 − 0.647i)9-s + (−0.548 + 0.950i)11-s − 0.281·13-s − 0.389i·15-s + (−0.150 − 0.0870i)17-s + (−1.25 + 0.723i)19-s + (0.143 − 0.481i)21-s + (−0.657 + 0.379i)23-s + (0.200 − 0.346i)25-s + 0.878i·27-s − 0.851i·29-s + (0.676 − 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.773 - 0.633i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.773 - 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227301 + 0.636785i\)
\(L(\frac12)\) \(\approx\) \(0.227301 + 0.636785i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.615 - 2.57i)T \)
good3 \( 1 + (0.753 + 0.435i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.81 - 3.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.01T + 13T^{2} \)
17 \( 1 + (0.621 + 0.358i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.45 - 3.15i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.15 - 1.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.58iT - 29T^{2} \)
31 \( 1 + (-3.76 + 6.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.27 - 3.62i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (1.30 + 2.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.82 - 2.20i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.753 + 0.435i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.27 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.88 - 4.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.5iT - 71T^{2} \)
73 \( 1 + (3.25 + 1.88i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.76 + 3.32i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.48iT - 83T^{2} \)
89 \( 1 + (6.98 - 4.03i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.01iT - 97T^{2} \)
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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

−10.21920695920437914110215214899, −9.881514270542123936344957900625, −8.703935205202934875292789862669, −7.960505817174477815169393574151, −6.78669362903122301024186663316, −6.18725360086711032737408861831, −5.40060391438344011902147491423, −4.25901364766853716237338647281, −2.79640512058846783119174452644, −1.95438408698329536367334159131, 0.31630382852758923476818300814, 1.94868232305762753789096521749, 3.41306470549678344930315730262, 4.73675365590227145019397530907, 5.14675980890371115217988870346, 6.24595447101838575520451939536, 7.19618182309425258051720729234, 8.361620609869771428016593811009, 8.716169733593861922451643837463, 10.03384499930856891125477548258

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