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Properties

Label 2-448-7.2-c3-0-7
Degree $2$
Conductor $448$
Sign $-0.0257 - 0.999i$
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  + (−1.40 − 2.42i)3-s + (−3.34 + 5.79i)5-s + (8.65 + 16.3i)7-s + (9.56 − 16.5i)9-s + (−15.7 − 27.2i)11-s − 18.6·13-s + 18.7·15-s + (43.9 + 76.1i)17-s + (−6.54 + 11.3i)19-s + (27.6 − 43.9i)21-s + (−4.68 + 8.12i)23-s + (40.1 + 69.4i)25-s − 129.·27-s + 5.11·29-s + (−128. − 222. i)31-s + ⋯
L(s)  = 1  + (−0.269 − 0.467i)3-s + (−0.299 + 0.518i)5-s + (0.467 + 0.884i)7-s + (0.354 − 0.613i)9-s + (−0.431 − 0.747i)11-s − 0.398·13-s + 0.323·15-s + (0.627 + 1.08i)17-s + (−0.0790 + 0.136i)19-s + (0.287 − 0.456i)21-s + (−0.0425 + 0.0736i)23-s + (0.320 + 0.555i)25-s − 0.922·27-s + 0.0327·29-s + (−0.745 − 1.29i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0257 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0257 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.0257 - 0.999i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.0257 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.137408051\)
\(L(\frac12)\) \(\approx\) \(1.137408051\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.65 - 16.3i)T \)
good3 \( 1 + (1.40 + 2.42i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (3.34 - 5.79i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (15.7 + 27.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 18.6T + 2.19e3T^{2} \)
17 \( 1 + (-43.9 - 76.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (6.54 - 11.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (4.68 - 8.12i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 5.11T + 2.43e4T^{2} \)
31 \( 1 + (128. + 222. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (190. - 329. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 217.T + 6.89e4T^{2} \)
43 \( 1 - 377.T + 7.95e4T^{2} \)
47 \( 1 + (178. - 309. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-382. - 661. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-225. - 390. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (87.0 - 150. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-248. - 430. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 350.T + 3.57e5T^{2} \)
73 \( 1 + (-531. - 920. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-280. + 485. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.10e3T + 5.71e5T^{2} \)
89 \( 1 + (603. - 1.04e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.44e3T + 9.12e5T^{2} \)
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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

−11.06083530221465495626958038263, −10.10776114562093712285043904973, −9.015928344724765402120341126795, −8.084437421284554253913036157918, −7.25837903413896991910942207574, −6.14540898216734031630531637909, −5.46217239955692824110361010936, −3.94968222838359842930880329879, −2.76733997367256703519928013769, −1.33231065856804155026994277071, 0.40006755869215680586745078516, 1.98490523202262177591767345824, 3.71404939814021404885937258479, 4.85583071094418033234712947182, 5.15566759841392399673303206354, 7.03751940866497681202619056475, 7.56225173407332705311305676958, 8.613344281797387240377689518087, 9.803944689356276497129228370138, 10.39941520678097839720205726228

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