Properties

Label 2-224-32.29-c1-0-13
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: $\chi_{224} (29, \cdot )$
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

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

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