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Properties

Label 2-1148-41.32-c1-0-5
Degree $2$
Conductor $1148$
Sign $-0.888 - 0.459i$
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  + (−0.197 + 0.197i)3-s + 3.12i·5-s + (0.707 − 0.707i)7-s + 2.92i·9-s + (−0.0441 + 0.0441i)11-s + (−3.40 + 3.40i)13-s + (−0.617 − 0.617i)15-s + (2.97 + 2.97i)17-s + (−5.33 − 5.33i)19-s + 0.279i·21-s − 0.780·23-s − 4.77·25-s + (−1.16 − 1.16i)27-s + (4.24 − 4.24i)29-s − 4.87·31-s + ⋯
L(s)  = 1  + (−0.113 + 0.113i)3-s + 1.39i·5-s + (0.267 − 0.267i)7-s + 0.974i·9-s + (−0.0133 + 0.0133i)11-s + (−0.944 + 0.944i)13-s + (−0.159 − 0.159i)15-s + (0.721 + 0.721i)17-s + (−1.22 − 1.22i)19-s + 0.0609i·21-s − 0.162·23-s − 0.954·25-s + (−0.225 − 0.225i)27-s + (0.788 − 0.788i)29-s − 0.875·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.888 - 0.459i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032067111\)
\(L(\frac12)\) \(\approx\) \(1.032067111\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.85 - 5.11i)T \)
good3 \( 1 + (0.197 - 0.197i)T - 3iT^{2} \)
5 \( 1 - 3.12iT - 5T^{2} \)
11 \( 1 + (0.0441 - 0.0441i)T - 11iT^{2} \)
13 \( 1 + (3.40 - 3.40i)T - 13iT^{2} \)
17 \( 1 + (-2.97 - 2.97i)T + 17iT^{2} \)
19 \( 1 + (5.33 + 5.33i)T + 19iT^{2} \)
23 \( 1 + 0.780T + 23T^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 + 4.87T + 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
43 \( 1 - 2.42iT - 43T^{2} \)
47 \( 1 + (4.29 + 4.29i)T + 47iT^{2} \)
53 \( 1 + (4.07 - 4.07i)T - 53iT^{2} \)
59 \( 1 + 5.85T + 59T^{2} \)
61 \( 1 - 8.77iT - 61T^{2} \)
67 \( 1 + (-7.81 - 7.81i)T + 67iT^{2} \)
71 \( 1 + (-6.22 + 6.22i)T - 71iT^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 + (5.78 - 5.78i)T - 79iT^{2} \)
83 \( 1 + 7.21T + 83T^{2} \)
89 \( 1 + (9.01 - 9.01i)T - 89iT^{2} \)
97 \( 1 + (-11.0 - 11.0i)T + 97iT^{2} \)
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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

−10.30092465591213488390171861825, −9.544062900302666256636140107209, −8.370267133039006683634674777748, −7.55194407249152377449781531692, −6.88056397098404200562705594931, −6.12119189632073066853681869035, −4.89405546593058143883317043364, −4.11877585045055062487159499922, −2.80109895266446428890465032601, −2.01039316771563759571611216002, 0.44060187159634593075066499724, 1.69509700447422953650294600374, 3.19721807873904336770794990155, 4.33724737193668621255762300145, 5.24364939472557745651300013725, 5.83044663803797763824647950700, 7.00490314889825373553279225371, 8.022866473473537690929628396973, 8.577263811713644139544667148003, 9.450349749859986464735823055981

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