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Properties

Label 2-224-32.29-c1-0-6
Degree $2$
Conductor $224$
Sign $0.880 - 0.473i$
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  + (−1.24 − 0.666i)2-s + (2.18 + 0.905i)3-s + (1.11 + 1.66i)4-s + (0.797 + 1.92i)5-s + (−2.12 − 2.58i)6-s + (0.707 − 0.707i)7-s + (−0.280 − 2.81i)8-s + (1.84 + 1.84i)9-s + (0.287 − 2.93i)10-s + (−1.24 + 0.513i)11-s + (0.927 + 4.64i)12-s + (−1.97 + 4.76i)13-s + (−1.35 + 0.411i)14-s + 4.93i·15-s + (−1.52 + 3.69i)16-s − 6.56i·17-s + ⋯
L(s)  = 1  + (−0.882 − 0.471i)2-s + (1.26 + 0.523i)3-s + (0.556 + 0.831i)4-s + (0.356 + 0.861i)5-s + (−0.867 − 1.05i)6-s + (0.267 − 0.267i)7-s + (−0.0991 − 0.995i)8-s + (0.613 + 0.613i)9-s + (0.0910 − 0.928i)10-s + (−0.374 + 0.154i)11-s + (0.267 + 1.34i)12-s + (−0.547 + 1.32i)13-s + (−0.361 + 0.109i)14-s + 1.27i·15-s + (−0.381 + 0.924i)16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.880 - 0.473i$
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: \((2,\ 224,\ (\ :1/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22288 + 0.307586i\)
\(L(\frac12)\) \(\approx\) \(1.22288 + 0.307586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.666i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-2.18 - 0.905i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.797 - 1.92i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (1.24 - 0.513i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.97 - 4.76i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 6.56iT - 17T^{2} \)
19 \( 1 + (-0.246 + 0.593i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.33 - 1.33i)T + 23iT^{2} \)
29 \( 1 + (3.76 + 1.55i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-0.633 - 1.52i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (2.76 + 2.76i)T + 41iT^{2} \)
43 \( 1 + (-0.855 + 0.354i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + (11.1 - 4.60i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-1.04 - 2.51i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (9.47 + 3.92i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (-9.85 - 4.08i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.76 - 6.76i)T - 71iT^{2} \)
73 \( 1 + (-0.141 - 0.141i)T + 73iT^{2} \)
79 \( 1 - 1.65iT - 79T^{2} \)
83 \( 1 + (-4.10 + 9.90i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.54 - 6.54i)T - 89iT^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−11.99671202439075347492136410647, −11.17029347076215699958512612288, −10.01376617880374105564310468278, −9.564851334895337594787042409833, −8.632638064379531053764937368536, −7.52782790446458616764675012604, −6.75630387515066317024077700397, −4.49202411187035793411469124946, −3.08562095706117250771009482205, −2.26294153983866956569949411812, 1.49385055441057418459700212018, 2.84918741308239052615735396350, 5.05495604989160525583616356625, 6.19638257917994629064158003900, 7.75661668428209728850518336119, 8.167226121881014488128269000072, 8.937586096962678260274255906165, 9.890731497853052991905643316062, 10.89997672881445154716388292966, 12.49998215553245037798427321448

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