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Properties

Label 2-1148-7.2-c1-0-0
Degree $2$
Conductor $1148$
Sign $-0.910 - 0.414i$
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  + (−1.05 − 1.83i)3-s + (−1.96 + 3.40i)5-s + (1.39 + 2.24i)7-s + (−0.742 + 1.28i)9-s + (−2.06 − 3.57i)11-s + 1.64·13-s + 8.33·15-s + (2.60 + 4.50i)17-s + (3.29 − 5.70i)19-s + (2.64 − 4.94i)21-s + (−3.58 + 6.21i)23-s + (−5.24 − 9.09i)25-s − 3.20·27-s − 9.38·29-s + (−3.23 − 5.59i)31-s + ⋯
L(s)  = 1  + (−0.611 − 1.05i)3-s + (−0.880 + 1.52i)5-s + (0.528 + 0.849i)7-s + (−0.247 + 0.428i)9-s + (−0.622 − 1.07i)11-s + 0.456·13-s + 2.15·15-s + (0.630 + 1.09i)17-s + (0.755 − 1.30i)19-s + (0.576 − 1.07i)21-s + (−0.747 + 1.29i)23-s + (−1.04 − 1.81i)25-s − 0.617·27-s − 1.74·29-s + (−0.580 − 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.910 - 0.414i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2085125664\)
\(L(\frac12)\) \(\approx\) \(0.2085125664\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 2.24i)T \)
41 \( 1 + T \)
good3 \( 1 + (1.05 + 1.83i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.96 - 3.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.06 + 3.57i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.64T + 13T^{2} \)
17 \( 1 + (-2.60 - 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.29 + 5.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.58 - 6.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.38T + 29T^{2} \)
31 \( 1 + (3.23 + 5.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.66 - 9.80i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 0.742T + 43T^{2} \)
47 \( 1 + (0.132 - 0.229i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.419 - 0.726i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.64 - 4.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.28 - 2.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.46 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.81T + 71T^{2} \)
73 \( 1 + (6.24 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.25 - 2.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 + (0.590 - 1.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.32T + 97T^{2} \)
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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

−10.50729950793333142325081053918, −9.275843334311430725300506957932, −8.043072127740066035474135952873, −7.72219317733746956617340583330, −6.87221730371173950176040609487, −5.97559365923791924005975521700, −5.50051815082875386641787123051, −3.74589317312373888334358246346, −2.98005073229810259305620004223, −1.69170748667557111250508554622, 0.10007100565998821532702557667, 1.57652499013126358119294292456, 3.80383901381286540314673676476, 4.18863010885503633787613760134, 5.21053125890608206669710470405, 5.38453943452038221049353293433, 7.29807833739248615189644897408, 7.75665432298035199979986495045, 8.673007075432118719129957075033, 9.613185068326541851243794850930

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