Properties

Label 2-224-224.37-c1-0-24
Degree $2$
Conductor $224$
Sign $-0.591 + 0.806i$
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  + (−0.399 − 1.35i)2-s + (1.62 − 2.11i)3-s + (−1.68 + 1.08i)4-s + (1.05 − 0.807i)5-s + (−3.51 − 1.35i)6-s + (2.64 − 0.0808i)7-s + (2.14 + 1.84i)8-s + (−1.05 − 3.94i)9-s + (−1.51 − 1.10i)10-s + (−1.69 + 0.223i)11-s + (−0.432 + 5.30i)12-s + (−0.550 + 0.228i)13-s + (−1.16 − 3.55i)14-s − 3.53i·15-s + (1.64 − 3.64i)16-s + (−4.66 + 2.69i)17-s + ⋯
L(s)  = 1  + (−0.282 − 0.959i)2-s + (0.935 − 1.21i)3-s + (−0.840 + 0.542i)4-s + (0.470 − 0.361i)5-s + (−1.43 − 0.552i)6-s + (0.999 − 0.0305i)7-s + (0.757 + 0.652i)8-s + (−0.352 − 1.31i)9-s + (−0.479 − 0.349i)10-s + (−0.511 + 0.0673i)11-s + (−0.124 + 1.53i)12-s + (−0.152 + 0.0632i)13-s + (−0.311 − 0.950i)14-s − 0.911i·15-s + (0.411 − 0.911i)16-s + (−1.13 + 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.591 + 0.806i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652957 - 1.28884i\)
\(L(\frac12)\) \(\approx\) \(0.652957 - 1.28884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.399 + 1.35i)T \)
7 \( 1 + (-2.64 + 0.0808i)T \)
good3 \( 1 + (-1.62 + 2.11i)T + (-0.776 - 2.89i)T^{2} \)
5 \( 1 + (-1.05 + 0.807i)T + (1.29 - 4.82i)T^{2} \)
11 \( 1 + (1.69 - 0.223i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 + (0.550 - 0.228i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (4.66 - 2.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.654 - 4.97i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (-1.35 - 5.05i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.87 + 9.36i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.43 - 2.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.83 + 3.71i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (-2.70 + 2.70i)T - 41iT^{2} \)
43 \( 1 + (-1.10 + 2.65i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (8.38 + 4.83i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.88 + 0.906i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (0.0283 + 0.215i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (-4.05 - 0.533i)T + (58.9 + 15.7i)T^{2} \)
67 \( 1 + (8.41 - 10.9i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-6.41 - 6.41i)T + 71iT^{2} \)
73 \( 1 + (9.61 + 2.57i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.05 - 2.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.11 + 3.77i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (16.1 - 4.31i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 9.28T + 97T^{2} \)
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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

−11.99645110464973450099301237585, −11.11614115752226312825131260707, −9.887902962604450548978294114683, −8.826773799504744874919775050136, −8.118026096412298494729224125780, −7.36822263797049273766178208278, −5.57391891074815623380634770767, −4.02637768829147126265110241828, −2.33374936606134708855517106420, −1.56654900866449681244176320733, 2.62158857276584649019378163987, 4.45052355869316055508580935177, 5.02971613096489616235053436469, 6.60100573681581003978105526252, 7.84645374257227040472399176946, 8.761053898497918863056556579881, 9.378033791748534246953257806651, 10.43894966809285565705820090433, 11.09328315545883291476732871249, 13.09321745359764774021278257939

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