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Properties

Label 2-224-32.13-c1-0-2
Degree $2$
Conductor $224$
Sign $0.235 - 0.971i$
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  + (−1.41 − 0.0115i)2-s + (−0.129 + 0.313i)3-s + (1.99 + 0.0327i)4-s + (−1.65 + 0.685i)5-s + (0.187 − 0.441i)6-s + (0.707 − 0.707i)7-s + (−2.82 − 0.0694i)8-s + (2.03 + 2.03i)9-s + (2.34 − 0.950i)10-s + (1.46 + 3.54i)11-s + (−0.269 + 0.622i)12-s + (−3.18 − 1.31i)13-s + (−1.00 + 0.991i)14-s − 0.608i·15-s + (3.99 + 0.130i)16-s + 5.24i·17-s + ⋯
L(s)  = 1  + (−0.999 − 0.00818i)2-s + (−0.0749 + 0.181i)3-s + (0.999 + 0.0163i)4-s + (−0.740 + 0.306i)5-s + (0.0764 − 0.180i)6-s + (0.267 − 0.267i)7-s + (−0.999 − 0.0245i)8-s + (0.679 + 0.679i)9-s + (0.742 − 0.300i)10-s + (0.442 + 1.06i)11-s + (−0.0779 + 0.179i)12-s + (−0.882 − 0.365i)13-s + (−0.269 + 0.265i)14-s − 0.157i·15-s + (0.999 + 0.0327i)16-s + 1.27i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529715 + 0.416880i\)
\(L(\frac12)\) \(\approx\) \(0.529715 + 0.416880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0115i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.129 - 0.313i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.65 - 0.685i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (-1.46 - 3.54i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.18 + 1.31i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 5.24iT - 17T^{2} \)
19 \( 1 + (-0.490 - 0.203i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-5.03 - 5.03i)T + 23iT^{2} \)
29 \( 1 + (1.56 - 3.76i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 7.16T + 31T^{2} \)
37 \( 1 + (-0.813 + 0.336i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (5.83 + 5.83i)T + 41iT^{2} \)
43 \( 1 + (-0.0662 - 0.159i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 + (4.75 + 11.4i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-11.2 + 4.64i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.04 - 4.94i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.461 - 1.11i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (0.00269 - 0.00269i)T - 71iT^{2} \)
73 \( 1 + (-2.92 - 2.92i)T + 73iT^{2} \)
79 \( 1 + 7.96iT - 79T^{2} \)
83 \( 1 + (6.66 + 2.76i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.79 - 2.79i)T - 89iT^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−12.15643201652121429419794873279, −11.32023770788148778989006290349, −10.34643440949824235101272678760, −9.772428713839357297429533317475, −8.414489583701788032760650080409, −7.45283494231335476668792491328, −6.93540486921475984146230616438, −5.12537203157789210999841639350, −3.67921423030455831836300220592, −1.82691976295309931830506841079, 0.819643404740775601678555730480, 2.87445120760820852139278729719, 4.56038538318944091262394926955, 6.24415231796349565778800637432, 7.18505355319716828505667229374, 8.152902349465165965862794399632, 9.091134292274302939922175224208, 9.887831910967790833970763552489, 11.26687104917174148437405462932, 11.80587442157087844738314514480

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