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Properties

Label 2-912-76.59-c1-0-9
Degree $2$
Conductor $912$
Sign $0.755 + 0.654i$
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 + (−0.826 + 0.300i)5-s + (0.907 − 0.524i)7-s + (0.173 + 0.984i)9-s + (3.79 + 2.19i)11-s + (−3.64 − 4.34i)13-s + (0.826 + 0.300i)15-s + (−0.539 + 3.05i)17-s + (4.28 − 0.788i)19-s + (−1.03 − 0.181i)21-s + (1.78 − 4.90i)23-s + (−3.23 + 2.71i)25-s + (0.500 − 0.866i)27-s + (6.30 − 1.11i)29-s + (3.40 + 5.88i)31-s + ⋯
L(s)  = 1  + (−0.442 − 0.371i)3-s + (−0.369 + 0.134i)5-s + (0.343 − 0.198i)7-s + (0.0578 + 0.328i)9-s + (1.14 + 0.661i)11-s + (−1.01 − 1.20i)13-s + (0.213 + 0.0776i)15-s + (−0.130 + 0.741i)17-s + (0.983 − 0.180i)19-s + (−0.225 − 0.0397i)21-s + (0.372 − 1.02i)23-s + (−0.647 + 0.543i)25-s + (0.0962 − 0.166i)27-s + (1.17 − 0.206i)29-s + (0.610 + 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.755 + 0.654i$
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: \((2,\ 912,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24091 - 0.462794i\)
\(L(\frac12)\) \(\approx\) \(1.24091 - 0.462794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.28 + 0.788i)T \)
good5 \( 1 + (0.826 - 0.300i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.907 + 0.524i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.79 - 2.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.64 + 4.34i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.539 - 3.05i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.78 + 4.90i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (-6.30 + 1.11i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.40 - 5.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.58iT - 37T^{2} \)
41 \( 1 + (-4.50 + 5.37i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.63 + 4.49i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-11.1 + 1.97i)T + (44.1 - 16.0i)T^{2} \)
53 \( 1 + (-1.83 + 5.02i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-2.62 + 14.8i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.56 - 2.75i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.709 - 4.02i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-3.59 + 1.30i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (10.4 + 8.76i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.54 - 2.97i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (6.02 - 3.47i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.71 - 6.81i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-3.33 - 0.587i)T + (91.1 + 33.1i)T^{2} \)
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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

−10.17380991097284197699014024914, −9.172699186620579073029137003079, −8.160579272530860400581541306547, −7.35184865799277143732269832042, −6.75369982589393033160579709264, −5.60903924608753212585917923939, −4.73709212038913243709730937534, −3.70572400550626737558793013265, −2.32748606940211903864412455800, −0.847222050758597136263426211324, 1.14357783312529414856199612189, 2.81243922636853451985235992483, 4.10970192531577293839517950489, 4.75322542716792886262385876807, 5.83054023506010667303871609028, 6.74451950594195293362338563547, 7.59458887379474259043394958256, 8.650610736106014005849462033786, 9.466650177546607493365647942090, 9.982851000538280708778844295580

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