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Properties

Label 2-448-112.27-c1-0-13
Degree $2$
Conductor $448$
Sign $0.315 + 0.948i$
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  + (1.73 − 1.73i)3-s + (1.73 − 1.73i)5-s + (2 − 1.73i)7-s − 2.99i·9-s + (−3 + 3i)11-s + (1.73 + 1.73i)13-s − 5.99i·15-s + (−5.19 + 5.19i)19-s + (0.464 − 6.46i)21-s − 4·23-s − 0.999i·25-s + (−1 + i)29-s + 6.92·31-s + 10.3i·33-s + (0.464 − 6.46i)35-s + ⋯
L(s)  = 1  + (0.999 − 0.999i)3-s + (0.774 − 0.774i)5-s + (0.755 − 0.654i)7-s − 0.999i·9-s + (−0.904 + 0.904i)11-s + (0.480 + 0.480i)13-s − 1.54i·15-s + (−1.19 + 1.19i)19-s + (0.101 − 1.41i)21-s − 0.834·23-s − 0.199i·25-s + (−0.185 + 0.185i)29-s + 1.24·31-s + 1.80i·33-s + (0.0784 − 1.09i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.315 + 0.948i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77043 - 1.27703i\)
\(L(\frac12)\) \(\approx\) \(1.77043 - 1.27703i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (-1.73 + 1.73i)T - 3iT^{2} \)
5 \( 1 + (-1.73 + 1.73i)T - 5iT^{2} \)
11 \( 1 + (3 - 3i)T - 11iT^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (5.19 - 5.19i)T - 19iT^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (1 - i)T - 29iT^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + (-1 + i)T - 43iT^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + (8.66 + 8.66i)T + 59iT^{2} \)
61 \( 1 + (5.19 + 5.19i)T + 61iT^{2} \)
67 \( 1 + (-7 - 7i)T + 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + (-1.73 + 1.73i)T - 83iT^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.75394406722260626457008333998, −9.960250891604774584576080773723, −8.851272935628582998623881583594, −8.150587815170368605984695726553, −7.49297242633907576214351225586, −6.39056139587399566686664815132, −5.13580793369108407138204949821, −4.00026884744127410785103692246, −2.19766122255295951016106398865, −1.55116467791809345414874933503, 2.36467818548282549620160120415, 3.01687137233733318075405017661, 4.39582372557744403637613722850, 5.51427358424172059383476477608, 6.47709431036068933829509773669, 8.097023630883514118928827000810, 8.533198746207018025616520850098, 9.458868589011985383933622960679, 10.55145719360837288237524214178, 10.72767159300261519967591279242

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