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Properties

Label 2-912-76.59-c1-0-18
Degree $2$
Conductor $912$
Sign $-0.483 + 0.875i$
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  + (−0.766 − 0.642i)3-s + (3.79 − 1.38i)5-s + (−0.699 + 0.404i)7-s + (0.173 + 0.984i)9-s + (−5.19 − 3.00i)11-s + (−3.41 − 4.06i)13-s + (−3.79 − 1.38i)15-s + (0.0141 − 0.0800i)17-s + (2.65 − 3.45i)19-s + (0.795 + 0.140i)21-s + (−0.391 + 1.07i)23-s + (8.68 − 7.28i)25-s + (0.500 − 0.866i)27-s + (−9.02 + 1.59i)29-s + (0.580 + 1.00i)31-s + ⋯
L(s)  = 1  + (−0.442 − 0.371i)3-s + (1.69 − 0.618i)5-s + (−0.264 + 0.152i)7-s + (0.0578 + 0.328i)9-s + (−1.56 − 0.905i)11-s + (−0.946 − 1.12i)13-s + (−0.980 − 0.356i)15-s + (0.00342 − 0.0194i)17-s + (0.609 − 0.792i)19-s + (0.173 + 0.0306i)21-s + (−0.0816 + 0.224i)23-s + (1.73 − 1.45i)25-s + (0.0962 − 0.166i)27-s + (−1.67 + 0.295i)29-s + (0.104 + 0.180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649443 - 1.10012i\)
\(L(\frac12)\) \(\approx\) \(0.649443 - 1.10012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-2.65 + 3.45i)T \)
good5 \( 1 + (-3.79 + 1.38i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.699 - 0.404i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.19 + 3.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 + 4.06i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.0141 + 0.0800i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (0.391 - 1.07i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (9.02 - 1.59i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.580 - 1.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.82iT - 37T^{2} \)
41 \( 1 + (-3.31 + 3.94i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.60 - 4.41i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-8.24 + 1.45i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-3.79 + 10.4i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.58 + 8.99i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.36 - 1.22i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.28 + 7.30i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-4.80 + 1.74i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-0.489 - 0.410i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (4.13 + 3.46i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (9.34 - 5.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.59 - 5.47i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (7.85 + 1.38i)T + (91.1 + 33.1i)T^{2} \)
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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

−9.856865567564724573680635302001, −9.140173548841116931321467024571, −8.113715183800849351371599595484, −7.25963689281768161019058664301, −6.05717060075261013567418604012, −5.38658297632778567310497887049, −5.10273627635123152886940530333, −2.99423953614605049545467130261, −2.16133439352575226933354767392, −0.59235359424277750582024045489, 1.94062803147906362909589896554, 2.69211245233958676929668984063, 4.26363736774763040988816971982, 5.38556798050952261556909192098, 5.81800870833540788913185417406, 6.96864023881738551591161063339, 7.52349737218273404522259127417, 9.129042119682912603434283193143, 9.780150675738655616256129920347, 10.18151061976433911813975812595

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