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Properties

Label 2-448-56.11-c2-0-30
Degree $2$
Conductor $448$
Sign $-0.527 + 0.849i$
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  + (2.75 − 4.76i)3-s + (4.79 − 2.76i)5-s + (4.79 − 5.10i)7-s + (−10.6 − 18.4i)9-s + (−7.54 + 13.0i)11-s − 11.8i·13-s − 30.4i·15-s + (2.62 − 4.54i)17-s + (14.5 + 25.2i)19-s + (−11.1 − 36.9i)21-s + (−4.42 + 2.55i)23-s + (2.81 − 4.88i)25-s − 67.9·27-s + 28.2i·29-s + (41.5 + 23.9i)31-s + ⋯
L(s)  = 1  + (0.917 − 1.58i)3-s + (0.958 − 0.553i)5-s + (0.684 − 0.728i)7-s + (−1.18 − 2.05i)9-s + (−0.686 + 1.18i)11-s − 0.912i·13-s − 2.03i·15-s + (0.154 − 0.267i)17-s + (0.765 + 1.32i)19-s + (−0.529 − 1.75i)21-s + (−0.192 + 0.111i)23-s + (0.112 − 0.195i)25-s − 2.51·27-s + 0.974i·29-s + (1.33 + 0.772i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.527 + 0.849i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.800770637\)
\(L(\frac12)\) \(\approx\) \(2.800770637\)
\(L(2)\) 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 \( 1 + (-4.79 + 5.10i)T \)
good3 \( 1 + (-2.75 + 4.76i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.79 + 2.76i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (7.54 - 13.0i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 11.8iT - 169T^{2} \)
17 \( 1 + (-2.62 + 4.54i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.5 - 25.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (4.42 - 2.55i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 28.2iT - 841T^{2} \)
31 \( 1 + (-41.5 - 23.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-27.0 + 15.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 10.3T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 + (14.4 - 8.34i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (16.1 + 9.34i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (34.4 - 59.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-18.4 + 10.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.93 - 17.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 127. iT - 5.04e3T^{2} \)
73 \( 1 + (9.54 - 16.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (3.20 - 1.84i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 5.08T + 6.88e3T^{2} \)
89 \( 1 + (51.5 + 89.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 62.3T + 9.40e3T^{2} \)
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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

−10.34557886829670838540815989620, −9.640342329084011490276237093575, −8.467612057098289032701430149264, −7.76532160870859975362713502630, −7.19900000936919949929200462598, −5.97594889976424885451427389710, −4.92753479075748199827530005964, −3.16611224058846963323032954396, −1.90299327680188399076507561457, −1.15170421377309745142451043601, 2.34297013708636923234711696500, 2.99829818480943211493341757889, 4.40641002781066326292096376564, 5.30205233598384568665191988009, 6.23975842383879233323483356535, 7.984935861820930861735256103050, 8.639205700461618088728543216844, 9.529857531093043026251221228545, 10.03877977288749011296437036024, 11.09277125725740219998406509616

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