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Properties

Label 2-448-112.83-c1-0-1
Degree $2$
Conductor $448$
Sign $0.418 - 0.908i$
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  + (−1.03 − 1.03i)3-s + (1.68 + 1.68i)5-s + (−1.38 + 2.25i)7-s − 0.851i·9-s + (2 + 2i)11-s + (−4.80 + 4.80i)13-s − 3.49i·15-s + 1.13i·17-s + (1.21 + 1.21i)19-s + (3.77 − 0.900i)21-s + 1.33·23-s + 0.690i·25-s + (−3.99 + 3.99i)27-s + (5.26 + 5.26i)29-s + 8.31·31-s + ⋯
L(s)  = 1  + (−0.598 − 0.598i)3-s + (0.754 + 0.754i)5-s + (−0.523 + 0.851i)7-s − 0.283i·9-s + (0.603 + 0.603i)11-s + (−1.33 + 1.33i)13-s − 0.902i·15-s + 0.275i·17-s + (0.279 + 0.279i)19-s + (0.823 − 0.196i)21-s + 0.278·23-s + 0.138i·25-s + (−0.768 + 0.768i)27-s + (0.978 + 0.978i)29-s + 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.418 - 0.908i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908997 + 0.581907i\)
\(L(\frac12)\) \(\approx\) \(0.908997 + 0.581907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 2.25i)T \)
good3 \( 1 + (1.03 + 1.03i)T + 3iT^{2} \)
5 \( 1 + (-1.68 - 1.68i)T + 5iT^{2} \)
11 \( 1 + (-2 - 2i)T + 11iT^{2} \)
13 \( 1 + (4.80 - 4.80i)T - 13iT^{2} \)
17 \( 1 - 1.13iT - 17T^{2} \)
19 \( 1 + (-1.21 - 1.21i)T + 19iT^{2} \)
23 \( 1 - 1.33T + 23T^{2} \)
29 \( 1 + (-5.26 - 5.26i)T + 29iT^{2} \)
31 \( 1 - 8.31T + 31T^{2} \)
37 \( 1 + (4.18 - 4.18i)T - 37iT^{2} \)
41 \( 1 - 1.63T + 41T^{2} \)
43 \( 1 + (-1.33 - 1.33i)T + 43iT^{2} \)
47 \( 1 + 1.93T + 47T^{2} \)
53 \( 1 + (-6.34 + 6.34i)T - 53iT^{2} \)
59 \( 1 + (3.29 - 3.29i)T - 59iT^{2} \)
61 \( 1 + (-2.04 + 2.04i)T - 61iT^{2} \)
67 \( 1 + (0.107 - 0.107i)T - 67iT^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 - 6.24T + 73T^{2} \)
79 \( 1 + 4.51iT - 79T^{2} \)
83 \( 1 + (9.71 + 9.71i)T + 83iT^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−11.59551721372427116551162655637, −10.15796222283668348901463053403, −9.644623613617397119648977400864, −8.719180448994576919492231133115, −7.04136552051618560309923062240, −6.69025814789463798738443698917, −5.89020003337855513736750366768, −4.64367891137456841342994497864, −2.94750294489168817563302175229, −1.79223958373391253588724227357, 0.73973947008576355208413646170, 2.79948552683761859489094832616, 4.32804587553929002657099665549, 5.17806146151722008062192439085, 5.96956663273328313372632507496, 7.19317674289999360233278126941, 8.254609723615819090124483190701, 9.482295572990808557642394595132, 10.03840962754119284679474097050, 10.70092062602853655201245361247

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