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Properties

Label 2-1148-41.32-c1-0-11
Degree $2$
Conductor $1148$
Sign $0.984 - 0.173i$
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  + (0.805 − 0.805i)3-s + 0.251i·5-s + (0.707 − 0.707i)7-s + 1.70i·9-s + (−2.93 + 2.93i)11-s + (1.89 − 1.89i)13-s + (0.202 + 0.202i)15-s + (2.33 + 2.33i)17-s + (1.33 + 1.33i)19-s − 1.13i·21-s + 6.67·23-s + 4.93·25-s + (3.78 + 3.78i)27-s + (−0.0155 + 0.0155i)29-s − 3.61·31-s + ⋯
L(s)  = 1  + (0.465 − 0.465i)3-s + 0.112i·5-s + (0.267 − 0.267i)7-s + 0.567i·9-s + (−0.886 + 0.886i)11-s + (0.524 − 0.524i)13-s + (0.0522 + 0.0522i)15-s + (0.566 + 0.566i)17-s + (0.307 + 0.307i)19-s − 0.248i·21-s + 1.39·23-s + 0.987·25-s + (0.729 + 0.729i)27-s + (−0.00288 + 0.00288i)29-s − 0.649·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.988014359\)
\(L(\frac12)\) \(\approx\) \(1.988014359\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.16 + 1.72i)T \)
good3 \( 1 + (-0.805 + 0.805i)T - 3iT^{2} \)
5 \( 1 - 0.251iT - 5T^{2} \)
11 \( 1 + (2.93 - 2.93i)T - 11iT^{2} \)
13 \( 1 + (-1.89 + 1.89i)T - 13iT^{2} \)
17 \( 1 + (-2.33 - 2.33i)T + 17iT^{2} \)
19 \( 1 + (-1.33 - 1.33i)T + 19iT^{2} \)
23 \( 1 - 6.67T + 23T^{2} \)
29 \( 1 + (0.0155 - 0.0155i)T - 29iT^{2} \)
31 \( 1 + 3.61T + 31T^{2} \)
37 \( 1 - 9.22T + 37T^{2} \)
43 \( 1 - 1.97iT - 43T^{2} \)
47 \( 1 + (2.41 + 2.41i)T + 47iT^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 + 4.91T + 59T^{2} \)
61 \( 1 + 9.53iT - 61T^{2} \)
67 \( 1 + (-8.59 - 8.59i)T + 67iT^{2} \)
71 \( 1 + (9.30 - 9.30i)T - 71iT^{2} \)
73 \( 1 + 5.52iT - 73T^{2} \)
79 \( 1 + (-4.22 + 4.22i)T - 79iT^{2} \)
83 \( 1 - 6.78T + 83T^{2} \)
89 \( 1 + (1.72 - 1.72i)T - 89iT^{2} \)
97 \( 1 + (2.76 + 2.76i)T + 97iT^{2} \)
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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

−9.932897400526939743683294831251, −8.833340259532516159329212931579, −8.022400131201230762780804265591, −7.52600546990487815835925784150, −6.70821890630327131882037513302, −5.43382710540975224447037260329, −4.78328580911136979733208265941, −3.43238921356898905469411423634, −2.47382259211329319473806232143, −1.30238060208913875996547950408, 0.980510308934807146839064277841, 2.75438597304766669663064899679, 3.38789138947175000962553460762, 4.60790030355503893318270343880, 5.41780868643795581956449061835, 6.39941975441774397589949083741, 7.39100499822609522819761547143, 8.361342839654425612368592959297, 8.988586078384469459967466926237, 9.561599335765187685603600240924

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