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Properties

Label 2-1456-13.4-c1-0-4
Degree $2$
Conductor $1456$
Sign $-0.661 - 0.750i$
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  + (0.126 − 0.218i)3-s + 1.14i·5-s + (0.866 − 0.5i)7-s + (1.46 + 2.54i)9-s + (−3.84 − 2.22i)11-s + (−3.54 + 0.666i)13-s + (0.251 + 0.145i)15-s + (1.35 + 2.34i)17-s + (−5.68 + 3.28i)19-s − 0.252i·21-s + (1.03 − 1.80i)23-s + 3.68·25-s + 1.50·27-s + (−3.59 + 6.23i)29-s + 7.90i·31-s + ⋯
L(s)  = 1  + (0.0729 − 0.126i)3-s + 0.513i·5-s + (0.327 − 0.188i)7-s + (0.489 + 0.847i)9-s + (−1.16 − 0.670i)11-s + (−0.982 + 0.184i)13-s + (0.0649 + 0.0374i)15-s + (0.328 + 0.569i)17-s + (−1.30 + 0.752i)19-s − 0.0551i·21-s + (0.216 − 0.375i)23-s + 0.736·25-s + 0.288·27-s + (−0.668 + 1.15i)29-s + 1.42i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-0.661 - 0.750i$
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: \((2,\ 1456,\ (\ :1/2),\ -0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8664626658\)
\(L(\frac12)\) \(\approx\) \(0.8664626658\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 - 0.666i)T \)
good3 \( 1 + (-0.126 + 0.218i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.14iT - 5T^{2} \)
11 \( 1 + (3.84 + 2.22i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.35 - 2.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.68 - 3.28i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.03 + 1.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.59 - 6.23i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.90iT - 31T^{2} \)
37 \( 1 + (8.35 + 4.82i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.22 + 4.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.70 - 2.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.67iT - 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + (0.0586 - 0.0338i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.05 + 7.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.444 - 0.256i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.34 - 5.39i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.02iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 15.3iT - 83T^{2} \)
89 \( 1 + (-9.40 - 5.42i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.84 + 1.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21389710778845974404522292599, −8.731418724999186675031002982817, −8.268740560629611099902349974218, −7.28914539269391671568648475076, −6.84093492634352706031806725052, −5.49890286458142238127037721891, −4.95525272769592164724602949829, −3.77705248964828749799197503825, −2.67713757214036314767655449881, −1.72831859406015553293609836352, 0.31941794727980061175936129415, 1.98031419375937382622907326605, 2.97276440907998953551765079341, 4.36589768466909127866020052205, 4.87700767604341756954769318554, 5.79971245786079014977467449968, 6.97799848494963519024451823257, 7.55137593070182581778753009077, 8.505430427274198619842017801302, 9.242402418430884218750928016356

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