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Properties

Label 2-1110-185.142-c1-0-30
Degree $2$
Conductor $1110$
Sign $0.867 + 0.498i$
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 + (2.21 + 0.314i)5-s + (0.707 − 0.707i)6-s + (−0.376 − 0.376i)7-s − i·8-s + 1.00i·9-s + (−0.314 + 2.21i)10-s − 1.28i·11-s + (0.707 + 0.707i)12-s − 5.12i·13-s + (0.376 − 0.376i)14-s + (−1.34 − 1.78i)15-s + 16-s − 2.31·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.990 + 0.140i)5-s + (0.288 − 0.288i)6-s + (−0.142 − 0.142i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.0994 + 0.700i)10-s − 0.387i·11-s + (0.204 + 0.204i)12-s − 1.42i·13-s + (0.100 − 0.100i)14-s + (−0.346 − 0.461i)15-s + 0.250·16-s − 0.560·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.867 + 0.498i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.867 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.412007760\)
\(L(\frac12)\) \(\approx\) \(1.412007760\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.314i)T \)
37 \( 1 + (-5.17 + 3.20i)T \)
good7 \( 1 + (0.376 + 0.376i)T + 7iT^{2} \)
11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 + 5.12iT - 13T^{2} \)
17 \( 1 + 2.31T + 17T^{2} \)
19 \( 1 + (-3.87 - 3.87i)T + 19iT^{2} \)
23 \( 1 + 6.29iT - 23T^{2} \)
29 \( 1 + (1.15 - 1.15i)T - 29iT^{2} \)
31 \( 1 + (7.11 + 7.11i)T + 31iT^{2} \)
41 \( 1 - 2.12iT - 41T^{2} \)
43 \( 1 + 2.56iT - 43T^{2} \)
47 \( 1 + (-2.64 - 2.64i)T + 47iT^{2} \)
53 \( 1 + (-0.572 + 0.572i)T - 53iT^{2} \)
59 \( 1 + (1.04 + 1.04i)T + 59iT^{2} \)
61 \( 1 + (-3.44 - 3.44i)T + 61iT^{2} \)
67 \( 1 + (-5.32 + 5.32i)T - 67iT^{2} \)
71 \( 1 - 4.38T + 71T^{2} \)
73 \( 1 + (-9.81 - 9.81i)T + 73iT^{2} \)
79 \( 1 + (11.3 + 11.3i)T + 79iT^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (-8.09 + 8.09i)T - 89iT^{2} \)
97 \( 1 - 8.07T + 97T^{2} \)
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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

−9.781979019841591237923316082555, −8.876658541247433015612227767559, −7.958407584473889401060928206443, −7.23763939800619433933959725747, −6.21744737687382063568461895008, −5.76686079510810868902436008724, −4.98097419326474437677844830049, −3.57490651125929325303373877852, −2.29095735820632380069943120291, −0.70520680952504279863003701665, 1.39936317621450217362367317968, 2.44715428359783785674309428813, 3.70335426766002465463441042134, 4.77297632082858154855205430734, 5.40791215546452749846607380361, 6.46570943758619874490080162908, 7.27133922500401998091720402651, 8.753581454689467657543726328364, 9.511576912138875678699210324134, 9.612904421249985600222309795896

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