↑

Properties

Label 2-896-16.5-c1-0-19
Degree $2$
Conductor $896$
Sign $0.0974 + 0.995i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.21 − 2.21i)3-s + (0.393 + 0.393i)5-s − i·7-s − 6.81i·9-s + (2.22 + 2.22i)11-s + (3.16 − 3.16i)13-s + 1.74·15-s + 0.980·17-s + (−5.26 + 5.26i)19-s + (−2.21 − 2.21i)21-s − 1.25i·23-s − 4.69i·25-s + (−8.46 − 8.46i)27-s + (−3.17 + 3.17i)29-s + 4.43·31-s + ⋯
L(s)  = 1  + (1.27 − 1.27i)3-s + (0.175 + 0.175i)5-s − 0.377i·7-s − 2.27i·9-s + (0.671 + 0.671i)11-s + (0.877 − 0.877i)13-s + 0.449·15-s + 0.237·17-s + (−1.20 + 1.20i)19-s + (−0.483 − 0.483i)21-s − 0.262i·23-s − 0.938i·25-s + (−1.62 − 1.62i)27-s + (−0.589 + 0.589i)29-s + 0.795·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0974 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0974 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.0974 + 0.995i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.0974 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89574 - 1.71917i\)
\(L(\frac12)\) \(\approx\) \(1.89574 - 1.71917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 2.21i)T - 3iT^{2} \)
5 \( 1 + (-0.393 - 0.393i)T + 5iT^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 - 0.980T + 17T^{2} \)
19 \( 1 + (5.26 - 5.26i)T - 19iT^{2} \)
23 \( 1 + 1.25iT - 23T^{2} \)
29 \( 1 + (3.17 - 3.17i)T - 29iT^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 + (0.645 + 0.645i)T + 37iT^{2} \)
41 \( 1 + 1.21iT - 41T^{2} \)
43 \( 1 + (-0.966 - 0.966i)T + 43iT^{2} \)
47 \( 1 + 9.97T + 47T^{2} \)
53 \( 1 + (-8.07 - 8.07i)T + 53iT^{2} \)
59 \( 1 + (-1.81 - 1.81i)T + 59iT^{2} \)
61 \( 1 + (-2.58 + 2.58i)T - 61iT^{2} \)
67 \( 1 + (1.59 - 1.59i)T - 67iT^{2} \)
71 \( 1 - 0.934iT - 71T^{2} \)
73 \( 1 + 0.710iT - 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (6.77 - 6.77i)T - 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 3.03T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.849560180036810035418899479254, −8.736739564439582855088994188891, −8.262716126863900431361266027485, −7.46955523156143372799554866927, −6.62750611108226015844400985188, −5.97816152365832097876769576792, −4.18469971858777626876586898167, −3.31547745282921532992495333269, −2.18166999731536237662284100839, −1.19554227974244012873455151436, 1.90270636904925593288939833675, 3.08679349513210041913664574866, 3.93561923011695494742866211197, 4.70189205725265124587335867308, 5.84905032208682583617542475280, 6.95249948998055301500573035986, 8.344173134164046731169433242552, 8.699290655507250489327340558602, 9.352305931837634753512080273285, 10.02424313224200244304139780857

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