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Properties

Label 2-1456-7.4-c1-0-34
Degree $2$
Conductor $1456$
Sign $0.605 + 0.795i$
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  + (1.11 − 1.93i)3-s + (1.11 + 1.93i)5-s + (2 − 1.73i)7-s + (−1 − 1.73i)9-s + (−1.5 + 2.59i)11-s − 13-s + 5.00·15-s + (3.73 − 6.47i)17-s + (1.5 + 2.59i)19-s + (−1.11 − 5.80i)21-s + (−1.88 − 3.25i)23-s + 2.23·27-s − 4.47·29-s + (2.5 − 4.33i)31-s + (3.35 + 5.80i)33-s + ⋯
L(s)  = 1  + (0.645 − 1.11i)3-s + (0.499 + 0.866i)5-s + (0.755 − 0.654i)7-s + (−0.333 − 0.577i)9-s + (−0.452 + 0.783i)11-s − 0.277·13-s + 1.29·15-s + (0.906 − 1.56i)17-s + (0.344 + 0.596i)19-s + (−0.243 − 1.26i)21-s + (−0.392 − 0.679i)23-s + 0.430·27-s − 0.830·29-s + (0.449 − 0.777i)31-s + (0.583 + 1.01i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.538998743\)
\(L(\frac12)\) \(\approx\) \(2.538998743\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
13 \( 1 + T \)
good3 \( 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.73 + 6.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.88 + 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.35 - 7.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-0.736 - 1.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.736 - 1.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (5.35 - 9.27i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.35 - 9.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.11 - 1.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−9.564222070422563519301169369152, −8.241519480061036594486697281862, −7.59278680409530293448056594533, −7.26892796926927244876551351508, −6.40050956864377328904812130976, −5.30308062608915446196897066752, −4.28260959373508114776295645087, −2.83814372702961403383215518368, −2.29478276037662563121514322592, −1.10725356519076593075455531569, 1.38130982358383683771444019109, 2.64838452845639695590210996746, 3.69926878954685653487473631342, 4.57082295022008393980249525195, 5.48383092411319898480191329049, 5.87209179820867116533439991561, 7.59397953468103056663730610355, 8.264332197645826148036507641986, 9.064856202699270087965394092742, 9.311457896694164501359503092680

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