Properties

Label 2-896-128.21-c1-0-4
Degree $2$
Conductor $896$
Sign $0.803 + 0.595i$
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  + (−0.372 − 1.36i)2-s + (−2.27 − 1.86i)3-s + (−1.72 + 1.01i)4-s + (−1.10 − 3.65i)5-s + (−1.70 + 3.80i)6-s + (−0.980 − 0.195i)7-s + (2.02 + 1.97i)8-s + (1.10 + 5.55i)9-s + (−4.56 + 2.87i)10-s + (−0.110 + 1.11i)11-s + (5.81 + 0.903i)12-s + (1.27 + 0.387i)13-s + (0.0992 + 1.41i)14-s + (−4.29 + 10.3i)15-s + (1.93 − 3.50i)16-s + (2.24 + 5.42i)17-s + ⋯
L(s)  = 1  + (−0.263 − 0.964i)2-s + (−1.31 − 1.07i)3-s + (−0.861 + 0.508i)4-s + (−0.495 − 1.63i)5-s + (−0.694 + 1.55i)6-s + (−0.370 − 0.0737i)7-s + (0.717 + 0.696i)8-s + (0.368 + 1.85i)9-s + (−1.44 + 0.907i)10-s + (−0.0332 + 0.337i)11-s + (1.67 + 0.260i)12-s + (0.354 + 0.107i)13-s + (0.0265 + 0.377i)14-s + (−1.10 + 2.67i)15-s + (0.483 − 0.875i)16-s + (0.545 + 1.31i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.803 + 0.595i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263000 - 0.0867842i\)
\(L(\frac12)\) \(\approx\) \(0.263000 - 0.0867842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.372 + 1.36i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
good3 \( 1 + (2.27 + 1.86i)T + (0.585 + 2.94i)T^{2} \)
5 \( 1 + (1.10 + 3.65i)T + (-4.15 + 2.77i)T^{2} \)
11 \( 1 + (0.110 - 1.11i)T + (-10.7 - 2.14i)T^{2} \)
13 \( 1 + (-1.27 - 0.387i)T + (10.8 + 7.22i)T^{2} \)
17 \( 1 + (-2.24 - 5.42i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (0.224 + 0.419i)T + (-10.5 + 15.7i)T^{2} \)
23 \( 1 + (0.848 - 1.26i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (0.259 - 0.0255i)T + (28.4 - 5.65i)T^{2} \)
31 \( 1 + (3.16 + 3.16i)T + 31iT^{2} \)
37 \( 1 + (-0.940 - 0.502i)T + (20.5 + 30.7i)T^{2} \)
41 \( 1 + (-5.73 - 3.83i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (7.06 - 5.80i)T + (8.38 - 42.1i)T^{2} \)
47 \( 1 + (9.80 - 4.06i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (10.4 + 1.03i)T + (51.9 + 10.3i)T^{2} \)
59 \( 1 + (-12.3 + 3.75i)T + (49.0 - 32.7i)T^{2} \)
61 \( 1 + (-6.88 + 8.39i)T + (-11.9 - 59.8i)T^{2} \)
67 \( 1 + (-0.404 + 0.492i)T + (-13.0 - 65.7i)T^{2} \)
71 \( 1 + (1.20 - 6.07i)T + (-65.5 - 27.1i)T^{2} \)
73 \( 1 + (-12.0 + 2.40i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (7.14 + 2.95i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (10.2 - 5.46i)T + (46.1 - 69.0i)T^{2} \)
89 \( 1 + (-1.60 - 2.40i)T + (-34.0 + 82.2i)T^{2} \)
97 \( 1 + (9.85 + 9.85i)T + 97iT^{2} \)
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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

−10.12139266267939661326888180266, −9.323324989332813679209871534162, −8.159346407440152555015349077080, −7.86753360698530908041379205637, −6.50084043420722911874322302698, −5.50268985918989001645975933434, −4.74519522599253216445207426264, −3.75625468523661047835594123361, −1.77816517617519212677367379833, −1.00575657059608162754747441737, 0.22584613855892977889218288820, 3.22083609421042467399093591821, 4.04769180984978951780141046440, 5.20607921744491211568049009995, 5.90973686741217323546653315238, 6.74029494308159200434738715692, 7.26385778546655983906004539080, 8.497762923157185482904514534317, 9.725359321900744606541457943558, 10.08376256161602655733079951679

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