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Properties

Label 2-448-7.5-c2-0-25
Degree $2$
Conductor $448$
Sign $0.868 + 0.495i$
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  + (3.45 − 1.99i)3-s + (7.80 + 4.50i)5-s + (5.54 − 4.26i)7-s + (3.44 − 5.96i)9-s + (−8.28 − 14.3i)11-s + 0.446i·13-s + 35.9·15-s + (6.02 − 3.47i)17-s + (−11.0 − 6.37i)19-s + (10.6 − 25.7i)21-s + (−13.2 + 23.0i)23-s + (28.1 + 48.7i)25-s + 8.43i·27-s − 26.4·29-s + (21.7 − 12.5i)31-s + ⋯
L(s)  = 1  + (1.15 − 0.664i)3-s + (1.56 + 0.901i)5-s + (0.792 − 0.609i)7-s + (0.382 − 0.662i)9-s + (−0.752 − 1.30i)11-s + 0.0343i·13-s + 2.39·15-s + (0.354 − 0.204i)17-s + (−0.580 − 0.335i)19-s + (0.507 − 1.22i)21-s + (−0.577 + 1.00i)23-s + (1.12 + 1.95i)25-s + 0.312i·27-s − 0.912·29-s + (0.702 − 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.868 + 0.495i$
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: \((2,\ 448,\ (\ :1),\ 0.868 + 0.495i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.347764997\)
\(L(\frac12)\) \(\approx\) \(3.347764997\)
\(L(2)\) 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 \( 1 + (-5.54 + 4.26i)T \)
good3 \( 1 + (-3.45 + 1.99i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-7.80 - 4.50i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.28 + 14.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 0.446iT - 169T^{2} \)
17 \( 1 + (-6.02 + 3.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.0 + 6.37i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (13.2 - 23.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 26.4T + 841T^{2} \)
31 \( 1 + (-21.7 + 12.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (31.6 - 54.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 0.519iT - 1.68e3T^{2} \)
43 \( 1 + 25.5T + 1.84e3T^{2} \)
47 \( 1 + (-59.4 - 34.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (3.58 + 6.20i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-65.3 + 37.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (39.8 + 23.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (21.4 + 37.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 60.0T + 5.04e3T^{2} \)
73 \( 1 + (40.5 - 23.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (27.1 - 47.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 11.4iT - 6.88e3T^{2} \)
89 \( 1 + (-53.1 - 30.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 20.3iT - 9.40e3T^{2} \)
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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

−10.67520243986864244537887950368, −9.941324296799489598954872635998, −8.898317847131345342923897441330, −8.038911883383393519722541092190, −7.26080668507333583814637107098, −6.20797155310412807629088214176, −5.26208325057690775322301469231, −3.40208468416246775692783813560, −2.49420117276846160082675152648, −1.51236667078505371441622674501, 1.87311494729622849864744048327, 2.44823188149261298188609606393, 4.25424136127537056964234502027, 5.10057971055895861669135901262, 5.97379025605603427968182442326, 7.58470728564165706394164095085, 8.699107845931986971439887104273, 8.947846649480569183396668042873, 10.10688167259958235962789635654, 10.32097179011549307827892289653

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