Properties

Label 2-1110-185.142-c1-0-9
Degree $2$
Conductor $1110$
Sign $0.990 - 0.138i$
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 + (1.50 + 1.65i)5-s + (−0.707 + 0.707i)6-s + (−1.06 − 1.06i)7-s + i·8-s + 1.00i·9-s + (1.65 − 1.50i)10-s + 4.34i·11-s + (0.707 + 0.707i)12-s − 4.56i·13-s + (−1.06 + 1.06i)14-s + (0.111 − 2.23i)15-s + 16-s − 3.41·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.670 + 0.741i)5-s + (−0.288 + 0.288i)6-s + (−0.401 − 0.401i)7-s + 0.353i·8-s + 0.333i·9-s + (0.524 − 0.474i)10-s + 1.31i·11-s + (0.204 + 0.204i)12-s − 1.26i·13-s + (−0.283 + 0.283i)14-s + (0.0287 − 0.576i)15-s + 0.250·16-s − 0.827·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.990 - 0.138i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.990 - 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.244356898\)
\(L(\frac12)\) \(\approx\) \(1.244356898\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 1.65i)T \)
37 \( 1 + (-2.01 - 5.73i)T \)
good7 \( 1 + (1.06 + 1.06i)T + 7iT^{2} \)
11 \( 1 - 4.34iT - 11T^{2} \)
13 \( 1 + 4.56iT - 13T^{2} \)
17 \( 1 + 3.41T + 17T^{2} \)
19 \( 1 + (-3.26 - 3.26i)T + 19iT^{2} \)
23 \( 1 - 3.95iT - 23T^{2} \)
29 \( 1 + (-1.85 + 1.85i)T - 29iT^{2} \)
31 \( 1 + (-3.73 - 3.73i)T + 31iT^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 + 2.82iT - 43T^{2} \)
47 \( 1 + (-5.10 - 5.10i)T + 47iT^{2} \)
53 \( 1 + (-4.58 + 4.58i)T - 53iT^{2} \)
59 \( 1 + (-2.79 - 2.79i)T + 59iT^{2} \)
61 \( 1 + (-1.42 - 1.42i)T + 61iT^{2} \)
67 \( 1 + (3.17 - 3.17i)T - 67iT^{2} \)
71 \( 1 - 9.34T + 71T^{2} \)
73 \( 1 + (6.35 + 6.35i)T + 73iT^{2} \)
79 \( 1 + (-0.205 - 0.205i)T + 79iT^{2} \)
83 \( 1 + (2.55 - 2.55i)T - 83iT^{2} \)
89 \( 1 + (-4.88 + 4.88i)T - 89iT^{2} \)
97 \( 1 + 7.85T + 97T^{2} \)
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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

−10.11009476891455455954565931171, −9.460194139214742727823872245342, −8.121966488010479909211104537830, −7.30991315805504084122119366842, −6.51897746669421762150006546013, −5.60218339849611807096948454519, −4.66089864145122735784301366335, −3.37704812407459215883265953342, −2.45957041376884339284650687912, −1.27836963371145879726177537213, 0.64333406521713671617451817683, 2.45942828363862562670822082589, 3.95477123595362966597083526925, 4.80762555004694885086251007207, 5.69919561163415592341132281284, 6.26949259101175681064329430929, 7.09703819293771877630375252319, 8.511948016016570352554974420455, 8.980595803229746834317737849572, 9.488895113200106462276134514893

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