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Properties

Label 2-896-112.109-c1-0-21
Degree $2$
Conductor $896$
Sign $0.999 + 0.00414i$
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  + (1.86 − 0.5i)3-s + (3.23 + 0.866i)5-s + (1.73 + 2i)7-s + (0.633 − 0.366i)9-s + (−1.13 − 4.23i)11-s + (−0.267 − 0.267i)13-s + 6.46·15-s + (0.232 − 0.401i)17-s + (1.13 − 4.23i)19-s + (4.23 + 2.86i)21-s + (−2.13 + 1.23i)23-s + (5.36 + 3.09i)25-s + (−3.09 + 3.09i)27-s + (3.73 + 3.73i)29-s + (−0.133 + 0.232i)31-s + ⋯
L(s)  = 1  + (1.07 − 0.288i)3-s + (1.44 + 0.387i)5-s + (0.654 + 0.755i)7-s + (0.211 − 0.122i)9-s + (−0.341 − 1.27i)11-s + (−0.0743 − 0.0743i)13-s + 1.66·15-s + (0.0562 − 0.0974i)17-s + (0.260 − 0.970i)19-s + (0.923 + 0.625i)21-s + (−0.444 + 0.256i)23-s + (1.07 + 0.619i)25-s + (−0.596 + 0.596i)27-s + (0.693 + 0.693i)29-s + (−0.0240 + 0.0416i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.999 + 0.00414i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.999 + 0.00414i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.89375 - 0.00600352i\)
\(L(\frac12)\) \(\approx\) \(2.89375 - 0.00600352i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (-1.86 + 0.5i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.23 - 0.866i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.13 + 4.23i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.267 + 0.267i)T + 13iT^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.13 + 4.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.13 - 1.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.73 - 3.73i)T + 29iT^{2} \)
31 \( 1 + (0.133 - 0.232i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.6 + 2.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.92iT - 41T^{2} \)
43 \( 1 + (-0.464 + 0.464i)T - 43iT^{2} \)
47 \( 1 + (3.86 + 6.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.96 - 11.0i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.66 + 9.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.0358 + 0.133i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-7.33 + 1.96i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + (2.76 + 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.330 + 0.571i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.46 + 8.46i)T + 83iT^{2} \)
89 \( 1 + (-4.5 + 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.983718023905754980378568028900, −9.012456853491825616151237909478, −8.656964399383306068096280779765, −7.77070845111386987936954978355, −6.66090363648024379078543751588, −5.71037531071381685834550188792, −5.08185438969279587249754500208, −3.22943742641066383537158788101, −2.58809341802531395305420415835, −1.65242267669658527500215050312, 1.63477265772981837911557444544, 2.36240929727067250972520998527, 3.76360735898957028571938824546, 4.74328952966191907842826477879, 5.62857309390917349857920309695, 6.78095526074120374856969763004, 7.78339381908563571498072123353, 8.495285658689733636087700964535, 9.375428917523433850921974099802, 10.09080756306800414376980707183

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