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Properties

Label 2-224-7.5-c2-0-3
Degree $2$
Conductor $224$
Sign $-0.726 - 0.687i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
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  + (−4.97 + 2.86i)3-s + (5.45 + 3.15i)5-s + (5.64 + 4.13i)7-s + (11.9 − 20.7i)9-s + (2.70 + 4.68i)11-s + 15.9i·13-s − 36.1·15-s + (−17.7 + 10.2i)17-s + (−11.7 − 6.79i)19-s + (−39.9 − 4.32i)21-s + (2.35 − 4.07i)23-s + (7.37 + 12.7i)25-s + 85.7i·27-s + 1.76·29-s + (11.9 − 6.87i)31-s + ⋯
L(s)  = 1  + (−1.65 + 0.956i)3-s + (1.09 + 0.630i)5-s + (0.807 + 0.590i)7-s + (1.33 − 2.30i)9-s + (0.245 + 0.426i)11-s + 1.22i·13-s − 2.41·15-s + (−1.04 + 0.602i)17-s + (−0.619 − 0.357i)19-s + (−1.90 − 0.206i)21-s + (0.102 − 0.177i)23-s + (0.294 + 0.510i)25-s + 3.17i·27-s + 0.0608·29-s + (0.383 − 0.221i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.374345 + 0.939600i\)
\(L(\frac12)\) \(\approx\) \(0.374345 + 0.939600i\)
\(L(2)\) 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 \( 1 + (-5.64 - 4.13i)T \)
good3 \( 1 + (4.97 - 2.86i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-5.45 - 3.15i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.70 - 4.68i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 15.9iT - 169T^{2} \)
17 \( 1 + (17.7 - 10.2i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.7 + 6.79i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.35 + 4.07i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 1.76T + 841T^{2} \)
31 \( 1 + (-11.9 + 6.87i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (5.23 - 9.06i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 11.2iT - 1.68e3T^{2} \)
43 \( 1 + 49.1T + 1.84e3T^{2} \)
47 \( 1 + (-9.02 - 5.20i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (16.0 + 27.8i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-57.2 + 33.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (27.6 + 15.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-49.2 - 85.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 61.7T + 5.04e3T^{2} \)
73 \( 1 + (-15.6 + 9.02i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-15.0 + 25.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 63.4iT - 6.88e3T^{2} \)
89 \( 1 + (-119. - 68.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 131. iT - 9.40e3T^{2} \)
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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

−11.97650758687940487284124777296, −11.32839718951799275685224756761, −10.58261600436101328639930938501, −9.747976719982404249483345197605, −8.862340711812098216647369296208, −6.69433849394940102417325499728, −6.25850075856689735304306710886, −5.08617169547158220571078190908, −4.26056025057735499813155612342, −1.93325317674117615646919904034, 0.69667925966916715460957959241, 1.85880789534000425918911589658, 4.71452665708206406374843582047, 5.48453897290925815049866507838, 6.33555719015994050565594294410, 7.39638116116358628539679400439, 8.528102930269887858428342323768, 10.10988142177099683765728248255, 10.84200799725992095191513607388, 11.64040977723014569763965086547

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