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Properties

Label 2-1148-41.9-c1-0-2
Degree $2$
Conductor $1148$
Sign $0.999 - 0.0429i$
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  + (−1.64 − 1.64i)3-s − 1.39i·5-s + (−0.707 − 0.707i)7-s + 2.39i·9-s + (3.79 + 3.79i)11-s + (3.40 + 3.40i)13-s + (−2.28 + 2.28i)15-s + (−2.83 + 2.83i)17-s + (−5.63 + 5.63i)19-s + 2.32i·21-s + 2.68·23-s + 3.06·25-s + (−0.990 + 0.990i)27-s + (−2.77 − 2.77i)29-s + 2.87·31-s + ⋯
L(s)  = 1  + (−0.948 − 0.948i)3-s − 0.622i·5-s + (−0.267 − 0.267i)7-s + 0.799i·9-s + (1.14 + 1.14i)11-s + (0.945 + 0.945i)13-s + (−0.590 + 0.590i)15-s + (−0.688 + 0.688i)17-s + (−1.29 + 1.29i)19-s + 0.506i·21-s + 0.560·23-s + 0.612·25-s + (−0.190 + 0.190i)27-s + (−0.515 − 0.515i)29-s + 0.516·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.999 - 0.0429i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.999 - 0.0429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.056549983\)
\(L(\frac12)\) \(\approx\) \(1.056549983\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.64 + 3.01i)T \)
good3 \( 1 + (1.64 + 1.64i)T + 3iT^{2} \)
5 \( 1 + 1.39iT - 5T^{2} \)
11 \( 1 + (-3.79 - 3.79i)T + 11iT^{2} \)
13 \( 1 + (-3.40 - 3.40i)T + 13iT^{2} \)
17 \( 1 + (2.83 - 2.83i)T - 17iT^{2} \)
19 \( 1 + (5.63 - 5.63i)T - 19iT^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 + (2.77 + 2.77i)T + 29iT^{2} \)
31 \( 1 - 2.87T + 31T^{2} \)
37 \( 1 - 6.03T + 37T^{2} \)
43 \( 1 - 1.35iT - 43T^{2} \)
47 \( 1 + (9.35 - 9.35i)T - 47iT^{2} \)
53 \( 1 + (-5.67 - 5.67i)T + 53iT^{2} \)
59 \( 1 - 6.26T + 59T^{2} \)
61 \( 1 + 8.87iT - 61T^{2} \)
67 \( 1 + (-3.51 + 3.51i)T - 67iT^{2} \)
71 \( 1 + (1.21 + 1.21i)T + 71iT^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (6.61 + 6.61i)T + 79iT^{2} \)
83 \( 1 + 3.94T + 83T^{2} \)
89 \( 1 + (0.812 + 0.812i)T + 89iT^{2} \)
97 \( 1 + (13.3 - 13.3i)T - 97iT^{2} \)
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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

−9.731597118789477628730945789513, −8.985735414516917750363371661486, −8.125495106497201699538382922942, −7.02308842054199460518276205475, −6.43564053067673000432549506829, −5.94510076303204446871611026757, −4.49579329726985772320273363131, −3.98528855929300234516578915370, −1.91950930955759019206805993221, −1.16629751969809462606566253192, 0.62263322023583688504729004786, 2.74247148999027818472947364853, 3.68965966780358980468273080724, 4.63432143994464811078883255252, 5.61246197789762782102830934744, 6.35062108671641745998303981026, 6.92906135183829211903740061646, 8.502091128675378093300301467419, 8.972620022300951925701283268508, 9.966471951135963001293304740954

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