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Properties

Label 2-448-7.5-c2-0-28
Degree $2$
Conductor $448$
Sign $-0.977 - 0.209i$
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  + (1.43 − 0.830i)3-s + (−7.27 − 4.20i)5-s + (3.99 − 5.74i)7-s + (−3.12 + 5.40i)9-s + (2.60 + 4.51i)11-s − 4.88i·13-s − 13.9·15-s + (6.68 − 3.86i)17-s + (−30.6 − 17.6i)19-s + (0.979 − 11.5i)21-s + (−13.3 + 23.0i)23-s + (22.7 + 39.4i)25-s + 25.3i·27-s − 45.1·29-s + (−35.0 + 20.2i)31-s + ⋯
L(s)  = 1  + (0.479 − 0.276i)3-s + (−1.45 − 0.840i)5-s + (0.571 − 0.820i)7-s + (−0.346 + 0.600i)9-s + (0.237 + 0.410i)11-s − 0.375i·13-s − 0.930·15-s + (0.393 − 0.227i)17-s + (−1.61 − 0.929i)19-s + (0.0466 − 0.551i)21-s + (−0.578 + 1.00i)23-s + (0.911 + 1.57i)25-s + 0.937i·27-s − 1.55·29-s + (−1.12 + 0.652i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.977 - 0.209i$
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: \((2,\ 448,\ (\ :1),\ -0.977 - 0.209i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4010558648\)
\(L(\frac12)\) \(\approx\) \(0.4010558648\)
\(L(2)\) 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 \( 1 + (-3.99 + 5.74i)T \)
good3 \( 1 + (-1.43 + 0.830i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (7.27 + 4.20i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.60 - 4.51i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 4.88iT - 169T^{2} \)
17 \( 1 + (-6.68 + 3.86i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (30.6 + 17.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (13.3 - 23.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 45.1T + 841T^{2} \)
31 \( 1 + (35.0 - 20.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-3.97 + 6.89i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 26.6iT - 1.68e3T^{2} \)
43 \( 1 + 0.403T + 1.84e3T^{2} \)
47 \( 1 + (28.2 + 16.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-40.6 - 70.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-7.62 + 4.40i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.3 - 14.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (43.7 + 75.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 27.5T + 5.04e3T^{2} \)
73 \( 1 + (-75.3 + 43.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-60.8 + 105. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 46.6iT - 6.88e3T^{2} \)
89 \( 1 + (52.7 + 30.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 66.4iT - 9.40e3T^{2} \)
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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

−10.70887988355551792693056100953, −9.201923991785489017833222210532, −8.428571313629351558308000803099, −7.66026221657067505007982959534, −7.19974319331491600221773228092, −5.35134383787069893963751886289, −4.40291168275959378608008405620, −3.56043121212941294178191273802, −1.78700237856870339438443010453, −0.15204883821650278857141968399, 2.29531796915920506570507593898, 3.59830164462801805134329085710, 4.16029468896077516862604947318, 5.82601396425609940560904577924, 6.76400757495414391123105770978, 8.083205403426607611170276751192, 8.364742242612701127578754857304, 9.438699404354485210472707934418, 10.67536931605912985148555819523, 11.40439211287664085855477054807

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