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Properties

Label 2-448-112.109-c1-0-4
Degree $2$
Conductor $448$
Sign $0.378 - 0.925i$
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  + (0.0543 − 0.0145i)3-s + (1.25 + 0.337i)5-s + (−0.230 + 2.63i)7-s + (−2.59 + 1.49i)9-s + (0.402 + 1.50i)11-s + (1.59 + 1.59i)13-s + 0.0733·15-s + (1.46 − 2.54i)17-s + (−2.05 + 7.65i)19-s + (0.0258 + 0.146i)21-s + (3.91 − 2.26i)23-s + (−2.86 − 1.65i)25-s + (−0.238 + 0.238i)27-s + (2.06 + 2.06i)29-s + (3.14 − 5.43i)31-s + ⋯
L(s)  = 1  + (0.0313 − 0.00841i)3-s + (0.562 + 0.150i)5-s + (−0.0872 + 0.996i)7-s + (−0.865 + 0.499i)9-s + (0.121 + 0.453i)11-s + (0.442 + 0.442i)13-s + 0.0189·15-s + (0.356 − 0.617i)17-s + (−0.470 + 1.75i)19-s + (0.00564 + 0.0320i)21-s + (0.816 − 0.471i)23-s + (−0.572 − 0.330i)25-s + (−0.0459 + 0.0459i)27-s + (0.382 + 0.382i)29-s + (0.564 − 0.976i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.378 - 0.925i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16017 + 0.779098i\)
\(L(\frac12)\) \(\approx\) \(1.16017 + 0.779098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.230 - 2.63i)T \)
good3 \( 1 + (-0.0543 + 0.0145i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.25 - 0.337i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.402 - 1.50i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \)
17 \( 1 + (-1.46 + 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.05 - 7.65i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.91 + 2.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 2.06i)T + 29iT^{2} \)
31 \( 1 + (-3.14 + 5.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + (-1.99 + 1.99i)T - 43iT^{2} \)
47 \( 1 + (0.979 + 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.00 - 11.2i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.793 + 2.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.60 + 9.72i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.11 + 0.566i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (12.2 + 7.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.961 - 1.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.82 + 8.82i)T + 83iT^{2} \)
89 \( 1 + (-11.5 + 6.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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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

−11.36920791921252272048517372897, −10.27933399437833315029988367827, −9.474041421761467883509920649436, −8.587201586206714419174128634445, −7.76804369886177934922162283384, −6.29199778795832142127258683378, −5.79179799498479529277664662727, −4.59682031197621602757501035244, −3.00840835912049952212169895857, −1.95274094742625830783970342797, 0.913751241003020627602320336292, 2.82607986582602909136330068322, 3.93724665807768352546479478549, 5.28414841606152329692172155747, 6.22695213749752950253732475765, 7.13294545872250519752169975139, 8.362340256723341320359231513310, 9.075371335996337752682405324205, 10.07129549790269213602797091472, 10.95279145871707395063051929754

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