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Properties

Label 2-1110-111.80-c1-0-36
Degree $2$
Conductor $1110$
Sign $0.249 + 0.968i$
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  + (0.707 + 0.707i)2-s + (−1.41 − i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−0.292 − 1.70i)6-s − 2·7-s + (−0.707 + 0.707i)8-s + (1.00 + 2.82i)9-s − 1.00·10-s + 2.82·11-s + (1.00 − 1.41i)12-s + (−3 − 3i)13-s + (−1.41 − 1.41i)14-s + (1.70 − 0.292i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.816 − 0.577i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.119 − 0.696i)6-s − 0.755·7-s + (−0.250 + 0.250i)8-s + (0.333 + 0.942i)9-s − 0.316·10-s + 0.852·11-s + (0.288 − 0.408i)12-s + (−0.832 − 0.832i)13-s + (−0.377 − 0.377i)14-s + (0.440 − 0.0756i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8294908337\)
\(L(\frac12)\) \(\approx\) \(0.8294908337\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (1.41 + i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (1 + 6i)T \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 + (-5.65 + 5.65i)T - 23iT^{2} \)
29 \( 1 + (1.41 + 1.41i)T + 29iT^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (-2.82 + 2.82i)T - 59iT^{2} \)
61 \( 1 + (-9 + 9i)T - 61iT^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + (-3 - 3i)T + 79iT^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 + (7.07 + 7.07i)T + 89iT^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−9.746554941722900351666356248318, −8.646006740115658257569201875200, −7.73391484278680266983849680968, −6.78443948156463303585026841876, −6.57025938650932784147372302505, −5.48945615930976143259219261948, −4.64052674023174438268446392028, −3.52932602153730475947145192373, −2.33798171180208863107156954478, −0.36730170171958952740006238815, 1.30838605387499848531414820031, 3.03642593667895516536287248582, 3.98183277996722052893011029523, 4.71843153264841544209102882447, 5.55487495827236575248859102969, 6.61455938537782027215786758742, 7.10353461244409363455700280572, 8.750166966124276048263673980221, 9.527854444805007611807361128238, 9.909370244106608897938006660032

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