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Properties

Label 2-224-32.21-c1-0-1
Degree $2$
Conductor $224$
Sign $-0.105 - 0.994i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 − 0.461i)2-s + (−0.825 + 0.341i)3-s + (1.57 + 1.23i)4-s + (−0.175 + 0.423i)5-s + (1.26 − 0.0761i)6-s + (0.707 + 0.707i)7-s + (−1.53 − 2.37i)8-s + (−1.55 + 1.55i)9-s + (0.430 − 0.485i)10-s + (−2.61 − 1.08i)11-s + (−1.72 − 0.479i)12-s + (0.791 + 1.91i)13-s + (−0.619 − 1.27i)14-s − 0.409i·15-s + (0.956 + 3.88i)16-s + 6.96i·17-s + ⋯
L(s)  = 1  + (−0.945 − 0.326i)2-s + (−0.476 + 0.197i)3-s + (0.787 + 0.616i)4-s + (−0.0785 + 0.189i)5-s + (0.514 − 0.0311i)6-s + (0.267 + 0.267i)7-s + (−0.542 − 0.839i)8-s + (−0.519 + 0.519i)9-s + (0.136 − 0.153i)10-s + (−0.788 − 0.326i)11-s + (−0.496 − 0.138i)12-s + (0.219 + 0.529i)13-s + (−0.165 − 0.339i)14-s − 0.105i·15-s + (0.239 + 0.971i)16-s + 1.68i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.105 - 0.994i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327573 + 0.364062i\)
\(L(\frac12)\) \(\approx\) \(0.327573 + 0.364062i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 0.461i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.825 - 0.341i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.175 - 0.423i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (2.61 + 1.08i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.791 - 1.91i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 6.96iT - 17T^{2} \)
19 \( 1 + (-0.0847 - 0.204i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (5.30 - 5.30i)T - 23iT^{2} \)
29 \( 1 + (-1.39 + 0.576i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.39T + 31T^{2} \)
37 \( 1 + (2.44 - 5.91i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-6.77 + 6.77i)T - 41iT^{2} \)
43 \( 1 + (3.54 + 1.46i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.928iT - 47T^{2} \)
53 \( 1 + (9.57 + 3.96i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.17 + 12.4i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-7.42 + 3.07i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.00 + 2.07i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.33 - 4.33i)T + 71iT^{2} \)
73 \( 1 + (-0.169 + 0.169i)T - 73iT^{2} \)
79 \( 1 + 0.467iT - 79T^{2} \)
83 \( 1 + (-4.32 - 10.4i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.43 - 2.43i)T + 89iT^{2} \)
97 \( 1 - 15.1T + 97T^{2} \)
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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

−12.18099558589320023756884594622, −11.19571515336671860124850556873, −10.75288469450199350820753894519, −9.737710118798261059149506781019, −8.466702369450089677449323971901, −7.899916342676598239262141669157, −6.46676103728827339737588448659, −5.40163096140881986387608947407, −3.61389084476065944389518091337, −2.00762638863621653659362484814, 0.56229444843784513381129139428, 2.68545998379923483278368840821, 4.90932798636008857470048586807, 5.98692722345255882970711262618, 7.06665986278145559699129798844, 8.018380047859861489340893414378, 8.977681741036074524632092810250, 10.06477269065798114439037843387, 10.92561776208714387196645536446, 11.79724053091730765666476428881

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