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Properties

Label 2-448-56.19-c1-0-4
Degree $2$
Conductor $448$
Sign $0.314 - 0.949i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
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  + (−0.182 + 0.105i)3-s + (−0.767 + 1.32i)5-s + (2.19 − 1.47i)7-s + (−1.47 + 2.55i)9-s + (2.37 + 4.11i)11-s − 4.95·13-s − 0.323i·15-s + (2.30 − 1.32i)17-s + (3.74 + 2.16i)19-s + (−0.244 + 0.500i)21-s + (0.547 + 0.316i)23-s + (1.32 + 2.29i)25-s − 1.25i·27-s + 7.85i·29-s + (0.645 + 1.11i)31-s + ⋯
L(s)  = 1  + (−0.105 + 0.0608i)3-s + (−0.343 + 0.594i)5-s + (0.829 − 0.558i)7-s + (−0.492 + 0.853i)9-s + (0.716 + 1.24i)11-s − 1.37·13-s − 0.0834i·15-s + (0.558 − 0.322i)17-s + (0.858 + 0.495i)19-s + (−0.0534 + 0.109i)21-s + (0.114 + 0.0659i)23-s + (0.264 + 0.458i)25-s − 0.241i·27-s + 1.45i·29-s + (0.115 + 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01494 + 0.733260i\)
\(L(\frac12)\) \(\approx\) \(1.01494 + 0.733260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 + 1.47i)T \)
good3 \( 1 + (0.182 - 0.105i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.767 - 1.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.95T + 13T^{2} \)
17 \( 1 + (-2.30 + 1.32i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.74 - 2.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.547 - 0.316i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.85iT - 29T^{2} \)
31 \( 1 + (-0.645 - 1.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 0.866i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.85iT - 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 + (-4.47 + 7.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.867 + 0.500i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.40 + 4.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.45 + 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.83 + 4.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (9.73 - 5.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.57 + 2.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 + (4.5 + 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.96iT - 97T^{2} \)
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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

−11.39317154174449613539320777176, −10.32923511500252892289205851298, −9.746731421810813072986167587080, −8.408557283363582756819947403307, −7.30091141836470877405671274533, −7.13142900627702490231572022877, −5.28591257090363879874105045724, −4.67156387165557698369396820483, −3.25633418164645562705961491466, −1.79637090688462409737021330489, 0.848279922490472356427378026928, 2.73136810755705438166544440353, 4.08028495865551222746364009839, 5.24877171479976810472364688994, 6.03600321361091727305067412857, 7.32481642319269094283676250175, 8.377433646326932198922077678959, 8.938614144245978174195965163174, 9.893749472061819554131682371367, 11.22394210158264783658882377717

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