Properties

Label 2-109-109.75-c1-0-5
Degree $2$
Conductor $109$
Sign $0.0148 + 0.999i$
Analytic cond. $0.870369$
Root an. cond. $0.932935$
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  + (1.33 − 2.31i)2-s + (1.04 + 0.874i)3-s + (−2.58 − 4.47i)4-s + (−0.127 + 0.725i)5-s + (3.42 − 1.24i)6-s + (−0.876 + 4.97i)7-s − 8.46·8-s + (−0.199 − 1.13i)9-s + (1.50 + 1.26i)10-s + (−2.15 + 1.80i)11-s + (1.21 − 6.91i)12-s + (0.730 − 4.14i)13-s + (10.3 + 8.68i)14-s + (−0.767 + 0.643i)15-s + (−6.16 + 10.6i)16-s + (0.913 − 1.58i)17-s + ⋯
L(s)  = 1  + (0.946 − 1.63i)2-s + (0.601 + 0.504i)3-s + (−1.29 − 2.23i)4-s + (−0.0571 + 0.324i)5-s + (1.39 − 0.508i)6-s + (−0.331 + 1.87i)7-s − 2.99·8-s + (−0.0665 − 0.377i)9-s + (0.477 + 0.400i)10-s + (−0.649 + 0.544i)11-s + (0.351 − 1.99i)12-s + (0.202 − 1.14i)13-s + (2.76 + 2.32i)14-s + (−0.198 + 0.166i)15-s + (−1.54 + 2.66i)16-s + (0.221 − 0.383i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0148 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0148 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(109\)
Sign: $0.0148 + 0.999i$
Analytic conductor: \(0.870369\)
Root analytic conductor: \(0.932935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 109,\ (\ :1/2),\ 0.0148 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15149 - 1.13453i\)
\(L(\frac12)\) \(\approx\) \(1.15149 - 1.13453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 + (1.62 - 10.3i)T \)
good2 \( 1 + (-1.33 + 2.31i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.04 - 0.874i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (0.127 - 0.725i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (0.876 - 4.97i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.15 - 1.80i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.730 + 4.14i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.913 + 1.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.447 + 0.774i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.13 + 1.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.71 + 3.11i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.777 - 4.40i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-1.56 - 8.90i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 - 4.99T + 41T^{2} \)
43 \( 1 + (-1.71 + 2.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.27 + 1.19i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-0.734 + 4.16i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.15 - 3.48i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (7.23 - 2.63i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.498 - 2.82i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.87 - 3.24i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.58 - 3.00i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-1.33 - 7.59i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.769 - 0.645i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.29 + 2.29i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-2.36 + 13.4i)T + (-91.1 - 33.1i)T^{2} \)
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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

−13.06591892192840896000086273206, −12.41011718140537761442697137699, −11.57565273803803333936075852350, −10.32411011317065630603375541025, −9.538818573891619639112856314764, −8.593514363316431139295921480544, −5.96754313618676548265331828899, −4.93323810586188426153266611144, −3.21271272712283681467359468444, −2.58872026793997817995313372485, 3.58669281144610423129116948799, 4.67403079686689877187160213974, 6.22634855107363987965759174609, 7.40332065413259579668929138265, 7.83070273667150364791950787791, 9.100076827840360451347709292119, 10.89077132129487087399779092652, 12.70352260959349132167256756258, 13.35913030915852627431481536918, 13.97588378982927235789556277154

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