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Properties

Label 2-1088-17.4-c1-0-15
Degree $2$
Conductor $1088$
Sign $0.994 - 0.100i$
Analytic cond. $8.68772$
Root an. cond. $2.94749$
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  + (2.08 − 2.08i)3-s + (−2.48 + 2.48i)5-s + (0.409 + 0.409i)7-s − 5.68i·9-s + (3.51 + 3.51i)11-s + 4.68·13-s + 10.3i·15-s + (−2.19 + 3.48i)17-s + 7.51i·19-s + 1.70·21-s + (−1.83 − 1.83i)23-s − 7.39i·25-s + (−5.59 − 5.59i)27-s + (0.196 − 0.196i)29-s + (5.18 − 5.18i)31-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)3-s + (−1.11 + 1.11i)5-s + (0.154 + 0.154i)7-s − 1.89i·9-s + (1.05 + 1.05i)11-s + 1.29·13-s + 2.67i·15-s + (−0.532 + 0.846i)17-s + 1.72i·19-s + 0.372·21-s + (−0.383 − 0.383i)23-s − 1.47i·25-s + (−1.07 − 1.07i)27-s + (0.0364 − 0.0364i)29-s + (0.931 − 0.931i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1088\)    =    \(2^{6} \cdot 17\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(8.68772\)
Root analytic conductor: \(2.94749\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1088} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1088,\ (\ :1/2),\ 0.994 - 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.191232236\)
\(L(\frac12)\) \(\approx\) \(2.191232236\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (2.19 - 3.48i)T \)
good3 \( 1 + (-2.08 + 2.08i)T - 3iT^{2} \)
5 \( 1 + (2.48 - 2.48i)T - 5iT^{2} \)
7 \( 1 + (-0.409 - 0.409i)T + 7iT^{2} \)
11 \( 1 + (-3.51 - 3.51i)T + 11iT^{2} \)
13 \( 1 - 4.68T + 13T^{2} \)
19 \( 1 - 7.51iT - 19T^{2} \)
23 \( 1 + (1.83 + 1.83i)T + 23iT^{2} \)
29 \( 1 + (-0.196 + 0.196i)T - 29iT^{2} \)
31 \( 1 + (-5.18 + 5.18i)T - 31iT^{2} \)
37 \( 1 + (-0.489 + 0.489i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
43 \( 1 + 2.52iT - 43T^{2} \)
47 \( 1 - 1.63T + 47T^{2} \)
53 \( 1 - 2.97iT - 53T^{2} \)
59 \( 1 + 7.02iT - 59T^{2} \)
61 \( 1 + (-4.48 - 4.48i)T + 61iT^{2} \)
67 \( 1 + 0.490T + 67T^{2} \)
71 \( 1 + (5.39 - 5.39i)T - 71iT^{2} \)
73 \( 1 + (-7.39 + 7.39i)T - 73iT^{2} \)
79 \( 1 + (-9.35 - 9.35i)T + 79iT^{2} \)
83 \( 1 + 1.30iT - 83T^{2} \)
89 \( 1 + 0.335T + 89T^{2} \)
97 \( 1 + (-1.29 + 1.29i)T - 97iT^{2} \)
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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

−9.787375594728272869125273173731, −8.666225811303762709387276837566, −8.127823764522141895898333899780, −7.54069578143112847679208549126, −6.62899450710438129209113906079, −6.23316016915907533044314963400, −3.93243292584741725758049968353, −3.76605377730726302040848386630, −2.42775163295639370908965442695, −1.47446896222924015458683307698, 0.982154001181287692564097265202, 2.92144738352802994083377855871, 3.77197721942265607445264186564, 4.36307637628206462958020291759, 5.11063371385195284648871369292, 6.56926212927194503889544321987, 7.80792067955791728343809711682, 8.541866337152588443751163746610, 8.939718460611708889380758610260, 9.402384147377735482200006639808

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