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Properties

Label 2-1122-17.13-c1-0-25
Degree $2$
Conductor $1122$
Sign $-0.237 + 0.971i$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
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 − i)5-s + (0.707 − 0.707i)6-s + (2.16 − 2.16i)7-s − i·8-s + 1.00i·9-s + (1 − i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + 0.982·13-s + (2.16 + 2.16i)14-s + 1.41i·15-s + 16-s + (−1.69 − 3.75i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.447 − 0.447i)5-s + (0.288 − 0.288i)6-s + (0.818 − 0.818i)7-s − 0.353i·8-s + 0.333i·9-s + (0.316 − 0.316i)10-s + (−0.213 + 0.213i)11-s + (0.204 + 0.204i)12-s + 0.272·13-s + (0.579 + 0.579i)14-s + 0.365i·15-s + 0.250·16-s + (−0.410 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7736498537\)
\(L(\frac12)\) \(\approx\) \(0.7736498537\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (1.69 + 3.75i)T \)
good5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 + (-2.16 + 2.16i)T - 7iT^{2} \)
13 \( 1 - 0.982T + 13T^{2} \)
19 \( 1 - 3.06iT - 19T^{2} \)
23 \( 1 + (0.897 - 0.897i)T - 23iT^{2} \)
29 \( 1 + (0.592 + 0.592i)T + 29iT^{2} \)
31 \( 1 + (3.09 + 3.09i)T + 31iT^{2} \)
37 \( 1 + (-0.472 - 0.472i)T + 37iT^{2} \)
41 \( 1 + (7.37 - 7.37i)T - 41iT^{2} \)
43 \( 1 + 8.78iT - 43T^{2} \)
47 \( 1 + 12.7T + 47T^{2} \)
53 \( 1 + 14.1iT - 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + (3.08 - 3.08i)T - 61iT^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 + (-1.18 - 1.18i)T + 71iT^{2} \)
73 \( 1 + (1.74 + 1.74i)T + 73iT^{2} \)
79 \( 1 + (-4.41 + 4.41i)T - 79iT^{2} \)
83 \( 1 + 2.29iT - 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 + (6.31 + 6.31i)T + 97iT^{2} \)
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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

−9.547941740496148472178939918006, −8.324676539407377947876779620943, −7.977767181062833631382234276868, −7.13168938552193695901029247113, −6.37848944317842088256264898916, −5.20059532405371602576153443536, −4.65306460241273727867857305596, −3.63846032700223724769093964102, −1.76838719650718236532746550268, −0.36640738519555625431972581604, 1.61384735585036166191699229834, 2.86246418753097351697183488294, 3.85838421090511800062477931106, 4.83439774865218689331698416116, 5.59839151339078615111776495347, 6.62577134136305608688570722622, 7.79724828369870685984810913046, 8.621782678985126382809565837215, 9.232693244189423029256994108998, 10.35677804719077851854050757645

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