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Properties

Label 2-224-28.23-c2-0-14
Degree $2$
Conductor $224$
Sign $0.237 + 0.971i$
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  + (3.51 − 2.02i)3-s + (1.21 − 2.10i)5-s + (−2.97 − 6.33i)7-s + (3.71 − 6.43i)9-s + (6.39 − 3.68i)11-s − 8.56·13-s − 9.86i·15-s + (−7.45 − 12.9i)17-s + (22.9 + 13.2i)19-s + (−23.2 − 16.2i)21-s + (12.9 + 7.45i)23-s + (9.54 + 16.5i)25-s + 6.35i·27-s − 52.0·29-s + (18.4 − 10.6i)31-s + ⋯
L(s)  = 1  + (1.17 − 0.675i)3-s + (0.243 − 0.421i)5-s + (−0.424 − 0.905i)7-s + (0.412 − 0.715i)9-s + (0.580 − 0.335i)11-s − 0.659·13-s − 0.657i·15-s + (−0.438 − 0.759i)17-s + (1.20 + 0.696i)19-s + (−1.10 − 0.772i)21-s + (0.561 + 0.324i)23-s + (0.381 + 0.660i)25-s + 0.235i·27-s − 1.79·29-s + (0.595 − 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.237 + 0.971i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.237 + 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73481 - 1.36115i\)
\(L(\frac12)\) \(\approx\) \(1.73481 - 1.36115i\)
\(L(2)\) 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 \( 1 + (2.97 + 6.33i)T \)
good3 \( 1 + (-3.51 + 2.02i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.21 + 2.10i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.39 + 3.68i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 8.56T + 169T^{2} \)
17 \( 1 + (7.45 + 12.9i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-22.9 - 13.2i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-12.9 - 7.45i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 52.0T + 841T^{2} \)
31 \( 1 + (-18.4 + 10.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-10.9 + 18.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 58.5T + 1.68e3T^{2} \)
43 \( 1 + 58.6iT - 1.84e3T^{2} \)
47 \( 1 + (-57.4 - 33.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-36.0 - 62.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (15.3 - 8.83i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (44.1 - 76.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (80.0 - 46.2i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 11.8iT - 5.04e3T^{2} \)
73 \( 1 + (33.2 + 57.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-15.9 - 9.19i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 101. iT - 6.88e3T^{2} \)
89 \( 1 + (23.2 - 40.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 135.T + 9.40e3T^{2} \)
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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

−12.04454697077208176727447695531, −10.82893640528274530068510441002, −9.408671731423383742496584598435, −9.096979547928814910401497755483, −7.52652312384203845862923836509, −7.27322662998839309749658855026, −5.63294044899319401002519351665, −4.00495615710315413574525906923, −2.79201539927769463267627367311, −1.18815922331347407384212677794, 2.34612763309079209074312743483, 3.27849913265425099870509849679, 4.62870768991178047835824649530, 6.09170095762860346636086041712, 7.30653778237182926527827209997, 8.587812603711316617887827104878, 9.346369007934803165856112879191, 9.906845528188327325727265962187, 11.18432074709983112091561925549, 12.30424878690031015385414856286

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