Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.999 - 0.0271i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 − 3.35i)3-s + (2.33 − 4.05i)5-s + (−6.95 − 0.792i)7-s + (−2.98 + 5.17i)9-s + (12.6 − 7.29i)11-s − 12.7·13-s − 18.1·15-s + (−16.9 + 9.76i)17-s + (−8.86 + 15.3i)19-s + (10.8 + 24.8i)21-s + (4.43 − 7.67i)23-s + (1.55 + 2.70i)25-s − 11.7·27-s − 35.4i·29-s + (−25.1 + 14.5i)31-s + ⋯
L(s)  = 1  + (−0.644 − 1.11i)3-s + (0.467 − 0.810i)5-s + (−0.993 − 0.113i)7-s + (−0.331 + 0.575i)9-s + (1.14 − 0.663i)11-s − 0.977·13-s − 1.20·15-s + (−0.994 + 0.574i)17-s + (−0.466 + 0.807i)19-s + (0.514 + 1.18i)21-s + (0.192 − 0.333i)23-s + (0.0623 + 0.108i)25-s − 0.433·27-s − 1.22i·29-s + (−0.810 + 0.468i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0271i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.999 - 0.0271i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.999 - 0.0271i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.0103434 + 0.762709i\)
\(L(\frac12)\)  \(\approx\)  \(0.0103434 + 0.762709i\)
\(L(2)\)   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 \( 1 + (6.95 + 0.792i)T \)
good3 \( 1 + (1.93 + 3.35i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.33 + 4.05i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-12.6 + 7.29i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (16.9 - 9.76i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (8.86 - 15.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-4.43 + 7.67i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 35.4iT - 841T^{2} \)
31 \( 1 + (25.1 - 14.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (10.5 + 6.10i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 22.0iT - 1.68e3T^{2} \)
43 \( 1 + 79.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.5 - 21.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-31.3 + 18.0i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-1.20 - 2.08i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.6 + 25.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (35.2 - 20.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 22.6T + 5.04e3T^{2} \)
73 \( 1 + (-66.1 + 38.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-68.4 + 118. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 49.9T + 6.88e3T^{2} \)
89 \( 1 + (-0.970 - 0.560i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 158. iT - 9.40e3T^{2} \)
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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

−11.94316579892678804844411849893, −10.66595776189791265522193058904, −9.451493874628593408527626031110, −8.662740620536418626291691580086, −7.19153605286662772340225594505, −6.39554197377761640082643548948, −5.60236072051738398149516709985, −3.95338708114531791686752255284, −1.92519523174617786973092882608, −0.43363270555767420681667953325, 2.59925132481842430250620115353, 4.05451734771947867148459221696, 5.12390164732891955702974622075, 6.47073168914522015839957284083, 7.05829935120340425520032616821, 9.218554937259922182406425610104, 9.594082424376198654960806923865, 10.54098585218650907115411172574, 11.30729362783031763247072396802, 12.36063274973249941146498821917

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