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Properties

Label 2-448-7.4-c3-0-25
Degree $2$
Conductor $448$
Sign $0.968 + 0.250i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.5 + 6.06i)3-s + (3.5 + 6.06i)5-s + (−14 − 12.1i)7-s + (−11 − 19.0i)9-s + (2.5 − 4.33i)11-s + 14·13-s − 49·15-s + (10.5 − 18.1i)17-s + (−24.5 − 42.4i)19-s + (122.5 − 42.4i)21-s + (−79.5 − 137. i)23-s + (38 − 65.8i)25-s − 35.0·27-s − 58·29-s + (73.5 − 127. i)31-s + ⋯
L(s)  = 1  + (−0.673 + 1.16i)3-s + (0.313 + 0.542i)5-s + (−0.755 − 0.654i)7-s + (−0.407 − 0.705i)9-s + (0.0685 − 0.118i)11-s + 0.298·13-s − 0.843·15-s + (0.149 − 0.259i)17-s + (−0.295 − 0.512i)19-s + (1.27 − 0.440i)21-s + (−0.720 − 1.24i)23-s + (0.303 − 0.526i)25-s − 0.249·27-s − 0.371·29-s + (0.425 − 0.737i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.060719711\)
\(L(\frac12)\) \(\approx\) \(1.060719711\)
\(L(\frac{5}{2})\) 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 \( 1 + (14 + 12.1i)T \)
good3 \( 1 + (3.5 - 6.06i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 14T + 2.19e3T^{2} \)
17 \( 1 + (-10.5 + 18.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (24.5 + 42.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (79.5 + 137. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 58T + 2.43e4T^{2} \)
31 \( 1 + (-73.5 + 127. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-109.5 - 189. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 350T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-262.5 - 454. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-151.5 + 262. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-52.5 + 90.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (206.5 + 357. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (207.5 - 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 432T + 3.57e5T^{2} \)
73 \( 1 + (-556.5 + 963. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (51.5 + 89.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 + (-164.5 - 284. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 882T + 9.12e5T^{2} \)
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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

−10.62240506139489001272366421856, −9.927167967635306365509112811152, −9.238251935205263441509814225402, −7.88513942124747585222291616700, −6.58506500854899234003332770020, −6.02190645106072940859777272613, −4.69581635357626105327492648608, −3.91933302504010352607203252584, −2.67423363470333866436314425923, −0.44827821665166528570442649629, 1.03785500779598773030214159615, 2.15094185677700939001104097232, 3.75264328317960321226236103452, 5.45420908597154314011653139094, 5.94806035131416538640983497923, 6.88895420754431933082961305050, 7.83310283940166509979952667173, 8.931151580329590855320822137884, 9.713611102826114979726388171354, 10.87855945724156514988503179153

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