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  :  $\chi_{224} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.345 - 0.938i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.501869 + 0.719220i\)
\(L(\frac12)\)  \(\approx\)  \(0.501869 + 0.719220i\)
\(L(2)\)   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

−12.27514051922719062943942393718, −11.56573772422508118159220547439, −10.41323946043259749365918568647, −9.371861938691979243716279488749, −8.385745880176002122304769446672, −7.30014547633501176239705985703, −6.09500388869881618694592716274, −5.14108806416473279743019041555, −3.59057708434199167407466643728, −1.99909742582060575444615696951, 0.47921383347889656192385013247, 2.75337086249543600104361649701, 4.30425626220740363114496079342, 5.33668011649849416936825617015, 6.60002348830032606953633535937, 7.948218520032679245314273661622, 8.469061744698170723466456572178, 10.07854155278945766736423490915, 10.85086459019500239952031775567, 11.43234194099895331850102812441

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