Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.345 - 0.938i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.729 − 0.421i)3-s + (−1.21 − 0.702i)5-s + (−1.56 + 6.82i)7-s + (−4.14 + 7.17i)9-s + (7.44 + 12.8i)11-s − 2.67i·13-s − 1.18·15-s + (11.7 − 6.77i)17-s + (25.2 + 14.5i)19-s + (1.73 + 5.63i)21-s + (−11.4 + 19.8i)23-s + (−11.5 − 19.9i)25-s + 14.5i·27-s − 3.76·29-s + (−11.4 + 6.63i)31-s + ⋯
L(s)  = 1  + (0.243 − 0.140i)3-s + (−0.243 − 0.140i)5-s + (−0.223 + 0.974i)7-s + (−0.460 + 0.797i)9-s + (0.676 + 1.17i)11-s − 0.205i·13-s − 0.0789·15-s + (0.690 − 0.398i)17-s + (1.32 + 0.767i)19-s + (0.0824 + 0.268i)21-s + (−0.497 + 0.861i)23-s + (−0.460 − 0.797i)25-s + 0.539i·27-s − 0.129·29-s + (−0.370 + 0.214i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.345 - 0.938i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.345 - 0.938i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.17950 + 0.823052i\)
\(L(\frac12)\)  \(\approx\)  \(1.17950 + 0.823052i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.56 - 6.82i)T \)
good3 \( 1 + (-0.729 + 0.421i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.21 + 0.702i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-7.44 - 12.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.67iT - 169T^{2} \)
17 \( 1 + (-11.7 + 6.77i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-25.2 - 14.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (11.4 - 19.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 3.76T + 841T^{2} \)
31 \( 1 + (11.4 - 6.63i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (1.32 - 2.29i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 45.2iT - 1.68e3T^{2} \)
43 \( 1 - 51.5T + 1.84e3T^{2} \)
47 \( 1 + (58.1 + 33.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (19.9 + 34.5i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-11.0 + 6.39i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-67.4 - 38.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.9 + 39.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 + (-55.6 + 32.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-40.2 + 69.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 121. iT - 6.88e3T^{2} \)
89 \( 1 + (39.7 + 22.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 134. iT - 9.40e3T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.04312904302355499580461889409, −11.61754176396278551967862030332, −10.04283174147070210260720002042, −9.360892642503607272105134907916, −8.157417857229938024168343129689, −7.40228253818132957442025080827, −5.92859306008131566958226437617, −4.94371983962777995805794678003, −3.33890539089660379450671329163, −1.89868251426243492415123981123, 0.804959339867129721082139697475, 3.21947728555198333097062358070, 3.97492706024230175200542313565, 5.70805717292695322652672690820, 6.77037119275219840658505154611, 7.84612727253916458341188948312, 8.980602925507061803418957395163, 9.790109511125513688261477406662, 11.01620453008022809675746612015, 11.66675684378755920935670583590

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