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Properties

Label 2-1134-21.20-c1-0-16
Degree $2$
Conductor $1134$
Sign $0.861 + 0.506i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  − i·2-s − 4-s + 1.79·5-s + (2.28 + 1.34i)7-s + i·8-s − 1.79i·10-s + 2.40i·11-s − 4.89i·13-s + (1.34 − 2.28i)14-s + 16-s + 3.66·17-s + 3.01i·19-s − 1.79·20-s + 2.40·22-s + 3.76i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.800·5-s + (0.861 + 0.506i)7-s + 0.353i·8-s − 0.566i·10-s + 0.724i·11-s − 1.35i·13-s + (0.358 − 0.609i)14-s + 0.250·16-s + 0.888·17-s + 0.692i·19-s − 0.400·20-s + 0.512·22-s + 0.785i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.861 + 0.506i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.861 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.006151952\)
\(L(\frac12)\) \(\approx\) \(2.006151952\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-2.28 - 1.34i)T \)
good5 \( 1 - 1.79T + 5T^{2} \)
11 \( 1 - 2.40iT - 11T^{2} \)
13 \( 1 + 4.89iT - 13T^{2} \)
17 \( 1 - 3.66T + 17T^{2} \)
19 \( 1 - 3.01iT - 19T^{2} \)
23 \( 1 - 3.76iT - 23T^{2} \)
29 \( 1 - 6.56iT - 29T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
37 \( 1 - 9.36T + 37T^{2} \)
41 \( 1 - 8.08T + 41T^{2} \)
43 \( 1 - 6.96T + 43T^{2} \)
47 \( 1 + 5.13T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 - 0.570T + 67T^{2} \)
71 \( 1 - 5.96iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + 3.74T + 89T^{2} \)
97 \( 1 + 5.51iT - 97T^{2} \)
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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

−9.690084028767756091707046894832, −9.301682600248394272659073470140, −7.967084640590606008972943074540, −7.69606835644865018870453876458, −5.98811925627827808347879918853, −5.49655181960804080757883530834, −4.57445914281695379593397597274, −3.30793730746271915850787244892, −2.25417618561486847018907915940, −1.28604922823270894401951735118, 1.10611197727195030109843867177, 2.47618831989350056195962651717, 4.04131041960535421513561566185, 4.76282771800891582947759938808, 5.82452671615489855782323529015, 6.42453936957718447046618170758, 7.44766737309465782842521769961, 8.129938399710131029676376671349, 9.083084625027708500773637051410, 9.648576849316772603455503509386

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