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Properties

Label 2-448-7.5-c2-0-27
Degree $2$
Conductor $448$
Sign $-0.232 + 0.972i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.97 − 2.86i)3-s + (−5.45 − 3.15i)5-s + (5.64 + 4.13i)7-s + (11.9 − 20.7i)9-s + (−2.70 − 4.68i)11-s − 15.9i·13-s − 36.1·15-s + (−17.7 + 10.2i)17-s + (11.7 + 6.79i)19-s + (39.9 + 4.32i)21-s + (2.35 − 4.07i)23-s + (7.37 + 12.7i)25-s − 85.7i·27-s − 1.76·29-s + (11.9 − 6.87i)31-s + ⋯
L(s)  = 1  + (1.65 − 0.956i)3-s + (−1.09 − 0.630i)5-s + (0.807 + 0.590i)7-s + (1.33 − 2.30i)9-s + (−0.245 − 0.426i)11-s − 1.22i·13-s − 2.41·15-s + (−1.04 + 0.602i)17-s + (0.619 + 0.357i)19-s + (1.90 + 0.206i)21-s + (0.102 − 0.177i)23-s + (0.294 + 0.510i)25-s − 3.17i·27-s − 0.0608·29-s + (0.383 − 0.221i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.232 + 0.972i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.232 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.594403059\)
\(L(\frac12)\) \(\approx\) \(2.594403059\)
\(L(2)\) 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 \( 1 + (-5.64 - 4.13i)T \)
good3 \( 1 + (-4.97 + 2.86i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (5.45 + 3.15i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.70 + 4.68i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 + (17.7 - 10.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.7 - 6.79i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.35 + 4.07i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 1.76T + 841T^{2} \)
31 \( 1 + (-11.9 + 6.87i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-5.23 + 9.06i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 11.2iT - 1.68e3T^{2} \)
43 \( 1 - 49.1T + 1.84e3T^{2} \)
47 \( 1 + (-9.02 - 5.20i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-16.0 - 27.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (57.2 - 33.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-27.6 - 15.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (49.2 + 85.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 61.7T + 5.04e3T^{2} \)
73 \( 1 + (-15.6 + 9.02i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-15.0 + 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 63.4iT - 6.88e3T^{2} \)
89 \( 1 + (-119. - 68.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 131. iT - 9.40e3T^{2} \)
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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

−10.67168578543134515882092105262, −9.197264301856894150761253364383, −8.573023492234599408970469066584, −7.932779420168792374146337556112, −7.53095403078693715014042971142, −6.03342785012711381408145696340, −4.51729245933165117166498357156, −3.42255073947195838929437912838, −2.34762516098056462576264798052, −0.939059488357489958597639427043, 2.09763219029818690031126480897, 3.25110341020708738617569594911, 4.25841555852326010658216193921, 4.71010802904678073225176599572, 7.15120314418269191777686093733, 7.48561619268796860384242289418, 8.505118853432061096435870698581, 9.215821307555332428759923072700, 10.17146145654596450254876164961, 11.05990268050977220101664613094

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