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Properties

Label 2-1110-185.142-c1-0-28
Degree $2$
Conductor $1110$
Sign $0.256 + 0.966i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−0.994 − 2.00i)5-s + (0.707 − 0.707i)6-s + (0.593 + 0.593i)7-s − i·8-s + 1.00i·9-s + (2.00 − 0.994i)10-s + 2.21i·11-s + (0.707 + 0.707i)12-s + 0.163i·13-s + (−0.593 + 0.593i)14-s + (−0.712 + 2.11i)15-s + 16-s + 5.23·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.444 − 0.895i)5-s + (0.288 − 0.288i)6-s + (0.224 + 0.224i)7-s − 0.353i·8-s + 0.333i·9-s + (0.633 − 0.314i)10-s + 0.668i·11-s + (0.204 + 0.204i)12-s + 0.0453i·13-s + (−0.158 + 0.158i)14-s + (−0.184 + 0.547i)15-s + 0.250·16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.256 + 0.966i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8823062602\)
\(L(\frac12)\) \(\approx\) \(0.8823062602\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.994 + 2.00i)T \)
37 \( 1 + (4.51 - 4.07i)T \)
good7 \( 1 + (-0.593 - 0.593i)T + 7iT^{2} \)
11 \( 1 - 2.21iT - 11T^{2} \)
13 \( 1 - 0.163iT - 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 + (1.67 + 1.67i)T + 19iT^{2} \)
23 \( 1 + 3.86iT - 23T^{2} \)
29 \( 1 + (-4.79 + 4.79i)T - 29iT^{2} \)
31 \( 1 + (5.64 + 5.64i)T + 31iT^{2} \)
41 \( 1 + 5.85iT - 41T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 + (7.97 + 7.97i)T + 47iT^{2} \)
53 \( 1 + (7.94 - 7.94i)T - 53iT^{2} \)
59 \( 1 + (-1.00 - 1.00i)T + 59iT^{2} \)
61 \( 1 + (7.50 + 7.50i)T + 61iT^{2} \)
67 \( 1 + (-5.39 + 5.39i)T - 67iT^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (-0.258 - 0.258i)T + 73iT^{2} \)
79 \( 1 + (1.08 + 1.08i)T + 79iT^{2} \)
83 \( 1 + (4.08 - 4.08i)T - 83iT^{2} \)
89 \( 1 + (-12.5 + 12.5i)T - 89iT^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.528122798457559551606533057017, −8.597767490415782815198628546001, −7.983981688358121978007976576548, −7.23890054819166100655486193685, −6.32393303975874271645177843973, −5.33436664862990649383628533114, −4.77080961950501955520801503483, −3.70955721604456376963452304158, −1.93864755809544202368009403902, −0.44834647852053899145717876986, 1.35708994110749659539122388065, 3.08569127325626231823559838240, 3.54685894025317131595918100300, 4.70147655882502939671691359937, 5.65953182565206329871738361962, 6.60020968015328420488104096968, 7.66592637191643377768893820451, 8.371199130021781268224717769445, 9.520102685190762042879536334953, 10.16669783902026004537032444936

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