Properties

Label 2-224-32.5-c1-0-18
Degree $2$
Conductor $224$
Sign $-0.109 + 0.994i$
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  + (1.19 − 0.759i)2-s + (−1.00 − 2.43i)3-s + (0.845 − 1.81i)4-s + (3.53 + 1.46i)5-s + (−3.05 − 2.13i)6-s + (0.707 + 0.707i)7-s + (−0.368 − 2.80i)8-s + (−2.78 + 2.78i)9-s + (5.32 − 0.938i)10-s + (−0.822 + 1.98i)11-s + (−5.26 − 0.230i)12-s + (−5.07 + 2.10i)13-s + (1.38 + 0.306i)14-s − 10.0i·15-s + (−2.57 − 3.06i)16-s + 4.22i·17-s + ⋯
L(s)  = 1  + (0.843 − 0.537i)2-s + (−0.581 − 1.40i)3-s + (0.422 − 0.906i)4-s + (1.57 + 0.654i)5-s + (−1.24 − 0.872i)6-s + (0.267 + 0.267i)7-s + (−0.130 − 0.991i)8-s + (−0.927 + 0.927i)9-s + (1.68 − 0.296i)10-s + (−0.247 + 0.598i)11-s + (−1.51 − 0.0664i)12-s + (−1.40 + 0.583i)13-s + (0.369 + 0.0818i)14-s − 2.59i·15-s + (−0.642 − 0.766i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.109 + 0.994i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.109 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27394 - 1.42148i\)
\(L(\frac12)\) \(\approx\) \(1.27394 - 1.42148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.759i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.00 + 2.43i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.53 - 1.46i)T + (3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.822 - 1.98i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (5.07 - 2.10i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 4.22iT - 17T^{2} \)
19 \( 1 + (3.28 - 1.36i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.42 + 3.42i)T - 23iT^{2} \)
29 \( 1 + (1.94 + 4.69i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 1.84T + 31T^{2} \)
37 \( 1 + (-6.71 - 2.77i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.82 + 1.82i)T - 41iT^{2} \)
43 \( 1 + (-1.21 + 2.92i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.88iT - 47T^{2} \)
53 \( 1 + (-1.83 + 4.43i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.50 - 1.03i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.93 + 11.9i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (3.16 + 7.65i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (6.98 + 6.98i)T + 71iT^{2} \)
73 \( 1 + (-6.18 + 6.18i)T - 73iT^{2} \)
79 \( 1 - 1.43iT - 79T^{2} \)
83 \( 1 + (4.24 - 1.75i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.46 + 5.46i)T + 89iT^{2} \)
97 \( 1 - 14.2T + 97T^{2} \)
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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

−12.37781037004648508155857847892, −11.21857398887549593449908910273, −10.33922874681441331695231351503, −9.425566942560835908940091037890, −7.50227775395962601570951855353, −6.48723339397346383260673008013, −5.99541839415746368814632935213, −4.81816790274664118019890219252, −2.40515059669593361727598257677, −1.85601183953948285392990853805, 2.74366090514417520855177549753, 4.46969155349740310801201775611, 5.27019589931114376761512303466, 5.71700286167658053634863210505, 7.21929104513453558475457733378, 8.832479189897646813118234827920, 9.652032374878021545981376932201, 10.57246142982423938140178761658, 11.50145673305156699537756069594, 12.76399111840441783960732973287

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