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Properties

Label 2-1456-13.4-c1-0-20
Degree $2$
Conductor $1456$
Sign $0.964 + 0.265i$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + (−1.36 + 2.36i)3-s − i·5-s + (−0.866 + 0.5i)7-s + (−2.23 − 3.86i)9-s + (−2.36 − 1.36i)11-s + (−2.59 + 2.5i)13-s + (2.36 + 1.36i)15-s + (−1.13 − 1.96i)17-s + (4.73 − 2.73i)19-s − 2.73i·21-s + (−2.36 + 4.09i)23-s + 4·25-s + 4.00·27-s + (1.5 − 2.59i)29-s − 8.73i·31-s + ⋯
L(s)  = 1  + (−0.788 + 1.36i)3-s − 0.447i·5-s + (−0.327 + 0.188i)7-s + (−0.744 − 1.28i)9-s + (−0.713 − 0.411i)11-s + (−0.720 + 0.693i)13-s + (0.610 + 0.352i)15-s + (−0.275 − 0.476i)17-s + (1.08 − 0.626i)19-s − 0.596i·21-s + (−0.493 + 0.854i)23-s + 0.800·25-s + 0.769·27-s + (0.278 − 0.482i)29-s − 1.56i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7884892081\)
\(L(\frac12)\) \(\approx\) \(0.7884892081\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (2.59 - 2.5i)T \)
good3 \( 1 + (1.36 - 2.36i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + iT - 5T^{2} \)
11 \( 1 + (2.36 + 1.36i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.13 + 1.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.73 + 2.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.73iT - 31T^{2} \)
37 \( 1 + (-4.5 - 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.86 + 2.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.901 - 1.56i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 - 8.46T + 53T^{2} \)
59 \( 1 + (6.29 - 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.13 - 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.90 + 2.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12 + 6.92i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.39iT - 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 14.1iT - 83T^{2} \)
89 \( 1 + (8.19 + 4.73i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.26 - 0.732i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.538903615849515440176073516629, −9.086503139617893631973032063038, −7.950286260010271987912334854133, −6.95543842419508850679310189977, −5.86902478667590904501740911032, −5.20256791801490244199555778025, −4.60764275361897195949598159279, −3.65146718210151160246675981654, −2.51160253385847862275957777955, −0.43334310756307661545481897136, 0.979692212261675132362058323426, 2.27480313829858430432388252056, 3.22213585606461087933432206654, 4.79153548224574181899063210725, 5.58851198027940288666663762912, 6.43845128141346341369842375305, 7.09219925780916484292324066627, 7.68239677059480595627933356748, 8.444009446513929806553026317051, 9.769771324659833845289942523286

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