Properties

Label 2-448-7.5-c2-0-11
Degree $2$
Conductor $448$
Sign $0.108 - 0.994i$
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  + (−4.19 + 2.41i)3-s + (0.0446 + 0.0257i)5-s + (6.12 + 3.39i)7-s + (7.20 − 12.4i)9-s + (−0.894 − 1.55i)11-s − 5.87i·13-s − 0.249·15-s + (23.0 − 13.2i)17-s + (22.8 + 13.1i)19-s + (−33.8 + 0.603i)21-s + (−12.8 + 22.2i)23-s + (−12.4 − 21.6i)25-s + 26.1i·27-s + 27.1·29-s + (−25.7 + 14.8i)31-s + ⋯
L(s)  = 1  + (−1.39 + 0.806i)3-s + (0.00892 + 0.00515i)5-s + (0.874 + 0.484i)7-s + (0.800 − 1.38i)9-s + (−0.0813 − 0.140i)11-s − 0.452i·13-s − 0.0166·15-s + (1.35 − 0.781i)17-s + (1.20 + 0.693i)19-s + (−1.61 + 0.0287i)21-s + (−0.558 + 0.966i)23-s + (−0.499 − 0.865i)25-s + 0.969i·27-s + 0.937·29-s + (−0.829 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.108 - 0.994i$
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: \((2,\ 448,\ (\ :1),\ 0.108 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.120465894\)
\(L(\frac12)\) \(\approx\) \(1.120465894\)
\(L(2)\) 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 \( 1 + (-6.12 - 3.39i)T \)
good3 \( 1 + (4.19 - 2.41i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-0.0446 - 0.0257i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (0.894 + 1.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 5.87iT - 169T^{2} \)
17 \( 1 + (-23.0 + 13.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-22.8 - 13.1i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (12.8 - 22.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 27.1T + 841T^{2} \)
31 \( 1 + (25.7 - 14.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (30.8 - 53.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 65.7iT - 1.68e3T^{2} \)
43 \( 1 + 9.52T + 1.84e3T^{2} \)
47 \( 1 + (-61.2 - 35.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.86 - 8.42i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-54.3 + 31.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-66.1 - 38.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.5 + 89.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 90.1T + 5.04e3T^{2} \)
73 \( 1 + (28.8 - 16.6i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (32.4 - 56.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 29.1iT - 6.88e3T^{2} \)
89 \( 1 + (-18.7 - 10.8i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 123. iT - 9.40e3T^{2} \)
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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

−11.21018738189768287242254397122, −10.14217889071938775009748928674, −9.759558748682660804364822598963, −8.334966344836926697134490328795, −7.41277249874377575680822812652, −5.94260943775371375446453245179, −5.42338354983395840988445178483, −4.63003093306972320449784048914, −3.24833437193241313517256754695, −1.15331925877848861674724393644, 0.70249484965947094230098591327, 1.86440027122723138025885839038, 3.95454389591663004065372333102, 5.24497349184444841964016492403, 5.78293445560339113842735709795, 7.13306312753678541949461313567, 7.48148173186864906385413205356, 8.743388787327962621170492762474, 10.18064216351976458397605912115, 10.78983120028155384228432867731

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