Properties

Label 2-1148-41.9-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.496 + 0.867i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (−0.796 − 0.796i)3-s − 1.53i·5-s + (0.707 + 0.707i)7-s − 1.73i·9-s + (2.10 + 2.10i)11-s + (3.47 + 3.47i)13-s + (−1.22 + 1.22i)15-s + (1.03 − 1.03i)17-s + (2.78 − 2.78i)19-s − 1.12i·21-s − 4.78·23-s + 2.62·25-s + (−3.76 + 3.76i)27-s + (1.87 + 1.87i)29-s + 1.64·31-s + ⋯
L(s)  = 1  + (−0.460 − 0.460i)3-s − 0.688i·5-s + (0.267 + 0.267i)7-s − 0.576i·9-s + (0.635 + 0.635i)11-s + (0.964 + 0.964i)13-s + (−0.316 + 0.316i)15-s + (0.250 − 0.250i)17-s + (0.638 − 0.638i)19-s − 0.245i·21-s − 0.998·23-s + 0.525·25-s + (−0.725 + 0.725i)27-s + (0.347 + 0.347i)29-s + 0.295·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.496 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542508321\)
\(L(\frac12)\) \(\approx\) \(1.542508321\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (5.27 + 3.63i)T \)
good3 \( 1 + (0.796 + 0.796i)T + 3iT^{2} \)
5 \( 1 + 1.53iT - 5T^{2} \)
11 \( 1 + (-2.10 - 2.10i)T + 11iT^{2} \)
13 \( 1 + (-3.47 - 3.47i)T + 13iT^{2} \)
17 \( 1 + (-1.03 + 1.03i)T - 17iT^{2} \)
19 \( 1 + (-2.78 + 2.78i)T - 19iT^{2} \)
23 \( 1 + 4.78T + 23T^{2} \)
29 \( 1 + (-1.87 - 1.87i)T + 29iT^{2} \)
31 \( 1 - 1.64T + 31T^{2} \)
37 \( 1 - 7.76T + 37T^{2} \)
43 \( 1 + 7.65iT - 43T^{2} \)
47 \( 1 + (-6.34 + 6.34i)T - 47iT^{2} \)
53 \( 1 + (3.97 + 3.97i)T + 53iT^{2} \)
59 \( 1 + 4.91T + 59T^{2} \)
61 \( 1 + 14.5iT - 61T^{2} \)
67 \( 1 + (11.3 - 11.3i)T - 67iT^{2} \)
71 \( 1 + (-6.65 - 6.65i)T + 71iT^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 + (-2.18 - 2.18i)T + 79iT^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 + (-11.2 - 11.2i)T + 89iT^{2} \)
97 \( 1 + (-12.3 + 12.3i)T - 97iT^{2} \)
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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

−9.338282946765510271560722242386, −9.056694414803823983908875865451, −8.057939917975784243919326108732, −6.97239719524545817020822536558, −6.41449479039926866917222888896, −5.44696135057167139241418741729, −4.51717390683476195615332653169, −3.57227132695960166230536419436, −1.91938429446615480135859620482, −0.895294003973373486423654544764, 1.20808901918651639494344306610, 2.85737517136193751794053137044, 3.78808367667669906040120212885, 4.75038313852995140820498394211, 5.93687482017770804373080092655, 6.23801531296872444265627552945, 7.71795288039483311647125164196, 8.057772058652577406493575680824, 9.230582066323432119291135928242, 10.28023546258442135521477177366

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