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Properties

Label 2-912-16.5-c1-0-46
Degree $2$
Conductor $912$
Sign $-0.271 + 0.962i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−0.0417 − 1.41i)2-s + (0.707 − 0.707i)3-s + (−1.99 + 0.118i)4-s + (1.69 + 1.69i)5-s + (−1.02 − 0.970i)6-s − 4.01i·7-s + (0.250 + 2.81i)8-s − 1.00i·9-s + (2.32 − 2.46i)10-s + (3.12 + 3.12i)11-s + (−1.32 + 1.49i)12-s + (3.31 − 3.31i)13-s + (−5.67 + 0.167i)14-s + 2.39·15-s + (3.97 − 0.471i)16-s + 1.28·17-s + ⋯
L(s)  = 1  + (−0.0295 − 0.999i)2-s + (0.408 − 0.408i)3-s + (−0.998 + 0.0590i)4-s + (0.758 + 0.758i)5-s + (−0.420 − 0.396i)6-s − 1.51i·7-s + (0.0884 + 0.996i)8-s − 0.333i·9-s + (0.735 − 0.780i)10-s + (0.942 + 0.942i)11-s + (−0.383 + 0.431i)12-s + (0.919 − 0.919i)13-s + (−1.51 + 0.0448i)14-s + 0.619·15-s + (0.993 − 0.117i)16-s + 0.311·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.271 + 0.962i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19258 - 1.57489i\)
\(L(\frac12)\) \(\approx\) \(1.19258 - 1.57489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0417 + 1.41i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
good5 \( 1 + (-1.69 - 1.69i)T + 5iT^{2} \)
7 \( 1 + 4.01iT - 7T^{2} \)
11 \( 1 + (-3.12 - 3.12i)T + 11iT^{2} \)
13 \( 1 + (-3.31 + 3.31i)T - 13iT^{2} \)
17 \( 1 - 1.28T + 17T^{2} \)
23 \( 1 - 8.95iT - 23T^{2} \)
29 \( 1 + (-3.81 + 3.81i)T - 29iT^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + (2.41 + 2.41i)T + 37iT^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 + (-0.432 - 0.432i)T + 43iT^{2} \)
47 \( 1 + 5.81T + 47T^{2} \)
53 \( 1 + (8.37 + 8.37i)T + 53iT^{2} \)
59 \( 1 + (-6.07 - 6.07i)T + 59iT^{2} \)
61 \( 1 + (-9.39 + 9.39i)T - 61iT^{2} \)
67 \( 1 + (4.53 - 4.53i)T - 67iT^{2} \)
71 \( 1 + 0.435iT - 71T^{2} \)
73 \( 1 - 6.52iT - 73T^{2} \)
79 \( 1 - 5.60T + 79T^{2} \)
83 \( 1 + (-1.87 + 1.87i)T - 83iT^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + 5.55T + 97T^{2} \)
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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

−9.969166740737748475512666719938, −9.384562722940765947785396557880, −8.188719228736642287709669034311, −7.37371058302715920301000076711, −6.53968556014545085555878081897, −5.38334201002252926953954504227, −3.91486454611499569098970889136, −3.48473218385067086692827293497, −2.06614877094834344850179923027, −1.10207998173910623414202005072, 1.48316881025925155050681443407, 3.10169703805566949845529190286, 4.37954826566835810320035489272, 5.21100841002514610836541820251, 6.12806401340478282590714436509, 6.54443672446895198029887294897, 8.264397172267445006960624899087, 8.773942754052837118037629822410, 9.054766605085649680892259758002, 9.856716443854540409895083932750

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