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Properties

Label 2-1148-41.18-c1-0-5
Degree $2$
Conductor $1148$
Sign $-0.345 - 0.938i$
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  + 2.07·3-s + (−2.15 + 1.56i)5-s + (−0.309 + 0.951i)7-s + 1.29·9-s + (0.978 + 0.711i)11-s + (1.50 + 4.63i)13-s + (−4.45 + 3.24i)15-s + (−4.06 − 2.95i)17-s + (−0.500 + 1.54i)19-s + (−0.640 + 1.97i)21-s + (1.09 + 3.38i)23-s + (0.639 − 1.96i)25-s − 3.52·27-s + (−0.0761 + 0.0553i)29-s + (−2.98 − 2.17i)31-s + ⋯
L(s)  = 1  + 1.19·3-s + (−0.961 + 0.698i)5-s + (−0.116 + 0.359i)7-s + 0.432·9-s + (0.295 + 0.214i)11-s + (0.417 + 1.28i)13-s + (−1.15 + 0.836i)15-s + (−0.985 − 0.716i)17-s + (−0.114 + 0.353i)19-s + (−0.139 + 0.430i)21-s + (0.229 + 0.705i)23-s + (0.127 − 0.393i)25-s − 0.678·27-s + (−0.0141 + 0.0102i)29-s + (−0.536 − 0.389i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.566086448\)
\(L(\frac12)\) \(\approx\) \(1.566086448\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-5.39 + 3.44i)T \)
good3 \( 1 - 2.07T + 3T^{2} \)
5 \( 1 + (2.15 - 1.56i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-0.978 - 0.711i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.50 - 4.63i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.06 + 2.95i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.500 - 1.54i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.09 - 3.38i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.0761 - 0.0553i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.98 + 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.15 - 4.46i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-3.06 - 9.43i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.59 - 7.99i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.13 + 3.00i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.23 + 6.88i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.09 - 6.45i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (11.8 - 8.63i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.09 - 6.60i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 4.58T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 0.731T + 83T^{2} \)
89 \( 1 + (-2.78 + 8.57i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-6.31 + 4.58i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.737122580705823113980501252623, −9.109958833935776316298613765991, −8.531451173369501627626491074144, −7.54397645001252552677745656129, −7.03706414209058599499095553843, −6.03284038726927749969761256097, −4.51869611507699966569096605098, −3.75016314410113959051774420292, −2.92944467405582009576528593117, −1.90718615467338512947025763685, 0.57049817027285142114836071546, 2.23409838659340827436115050564, 3.47394390401650474452639834514, 3.96772580560043506836796124608, 5.07256273163572404794733787974, 6.28337502582597176481197157149, 7.42513068171237439548575399179, 8.033676265149727109275252553315, 8.806336859879002379237397856201, 9.042225073894846191882931011285

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