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Properties

Label 2-224-32.21-c1-0-7
Degree $2$
Conductor $224$
Sign $-0.317 - 0.948i$
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  + (1.35 + 0.402i)2-s + (−1.81 + 0.752i)3-s + (1.67 + 1.09i)4-s + (−0.920 + 2.22i)5-s + (−2.76 + 0.288i)6-s + (−0.707 − 0.707i)7-s + (1.83 + 2.15i)8-s + (0.614 − 0.614i)9-s + (−2.14 + 2.64i)10-s + (−1.08 − 0.451i)11-s + (−3.86 − 0.723i)12-s + (1.57 + 3.79i)13-s + (−0.673 − 1.24i)14-s − 4.73i·15-s + (1.61 + 3.65i)16-s − 1.67i·17-s + ⋯
L(s)  = 1  + (0.958 + 0.284i)2-s + (−1.04 + 0.434i)3-s + (0.837 + 0.546i)4-s + (−0.411 + 0.994i)5-s + (−1.12 + 0.117i)6-s + (−0.267 − 0.267i)7-s + (0.647 + 0.762i)8-s + (0.204 − 0.204i)9-s + (−0.677 + 0.835i)10-s + (−0.328 − 0.136i)11-s + (−1.11 − 0.208i)12-s + (0.435 + 1.05i)13-s + (−0.180 − 0.332i)14-s − 1.22i·15-s + (0.403 + 0.914i)16-s − 0.406i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.317 - 0.948i$
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: \((2,\ 224,\ (\ :1/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802173 + 1.11434i\)
\(L(\frac12)\) \(\approx\) \(0.802173 + 1.11434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.402i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.81 - 0.752i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.920 - 2.22i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (1.08 + 0.451i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.57 - 3.79i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.67iT - 17T^{2} \)
19 \( 1 + (-0.573 - 1.38i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-4.84 + 4.84i)T - 23iT^{2} \)
29 \( 1 + (-8.76 + 3.62i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 2.81T + 31T^{2} \)
37 \( 1 + (2.45 - 5.93i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.01 + 2.01i)T - 41iT^{2} \)
43 \( 1 + (0.339 + 0.140i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.90iT - 47T^{2} \)
53 \( 1 + (9.67 + 4.00i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (2.01 - 4.86i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-13.2 + 5.47i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (11.5 - 4.77i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.25 - 7.25i)T + 71iT^{2} \)
73 \( 1 + (-0.621 + 0.621i)T - 73iT^{2} \)
79 \( 1 - 3.10iT - 79T^{2} \)
83 \( 1 + (-4.61 - 11.1i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.896 + 0.896i)T + 89iT^{2} \)
97 \( 1 + 9.63T + 97T^{2} \)
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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

−12.37560891386243094823585836859, −11.49177148791050744111530629439, −10.97565501234537714431354110959, −10.14551593499136431843758959178, −8.345648560860184879305385973232, −6.93763196764690105948479840372, −6.46376642093054073033372092902, −5.20174371722929312964119541190, −4.17137833186062364512504826011, −2.86773320674834284043599926090, 1.03520832670068030833559180287, 3.21670518976596762907527836699, 4.81244193085408410484035223348, 5.51825806717269550687917464313, 6.48355352103703427298591436332, 7.70875665567695593631560708858, 9.070151482862536268284605163013, 10.53758159637275228308556283764, 11.24810047039316640607414808143, 12.29253308730403739625087360414

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