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Properties

Label 2-448-112.69-c0-0-0
Degree $2$
Conductor $448$
Sign $0.923 + 0.382i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·7-s − i·9-s + (1 + i)11-s + i·25-s + (−1 + i)29-s + (−1 − i)37-s + (−1 − i)43-s − 49-s + (1 + i)53-s − 63-s + (−1 + i)67-s + 2i·71-s + (1 − i)77-s − 81-s + (1 − i)99-s + ⋯
L(s)  = 1  − i·7-s − i·9-s + (1 + i)11-s + i·25-s + (−1 + i)29-s + (−1 − i)37-s + (−1 − i)43-s − 49-s + (1 + i)53-s − 63-s + (−1 + i)67-s + 2i·71-s + (1 − i)77-s − 81-s + (1 − i)99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8788556582\)
\(L(\frac12)\) \(\approx\) \(0.8788556582\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + iT^{2} \)
5 \( 1 - iT^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.28453849739738996049913712636, −10.29676061488334960397887815207, −9.474016314938493682713111387006, −8.741116334776820343815578132825, −7.15390199970132659411349154919, −7.00291672716090395172122455978, −5.59279624830757051015126000143, −4.23739463083575739851460564498, −3.50179385784055813431133562100, −1.51577420840759217357557168397, 1.98122159041581201698815890823, 3.30998008914652883245525121937, 4.69881068639797943744084297627, 5.75676923305405776818062078547, 6.56223349854719080291047130507, 7.949616212203672260157876602020, 8.601569094420962261274312203669, 9.498060544688162463117669866862, 10.53743902176011287950925936075, 11.51476265427998245659871050178

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