Properties

Label 2-927-927.77-c1-0-13
Degree $2$
Conductor $927$
Sign $-0.335 - 0.942i$
Analytic cond. $7.40213$
Root an. cond. $2.72068$
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  + (1.38 − 0.587i)2-s + (−1.72 + 0.203i)3-s + (0.191 − 0.197i)4-s + (0.430 − 0.304i)5-s + (−2.26 + 1.29i)6-s + (−0.995 + 0.385i)7-s + (−0.940 + 2.42i)8-s + (2.91 − 0.698i)9-s + (0.418 − 0.676i)10-s + (0.481 − 0.637i)11-s + (−0.288 + 0.378i)12-s + (0.00532 − 0.0342i)13-s + (−1.15 + 1.12i)14-s + (−0.678 + 0.611i)15-s + (0.137 + 4.47i)16-s + (−5.38 − 1.89i)17-s + ⋯
L(s)  = 1  + (0.982 − 0.415i)2-s + (−0.993 + 0.117i)3-s + (0.0956 − 0.0985i)4-s + (0.192 − 0.136i)5-s + (−0.926 + 0.527i)6-s + (−0.376 + 0.145i)7-s + (−0.332 + 0.857i)8-s + (0.972 − 0.232i)9-s + (0.132 − 0.213i)10-s + (0.145 − 0.192i)11-s + (−0.0833 + 0.109i)12-s + (0.00147 − 0.00950i)13-s + (−0.308 + 0.299i)14-s + (−0.175 + 0.157i)15-s + (0.0344 + 1.11i)16-s + (−1.30 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.335 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.335 - 0.942i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.335 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520100 + 0.736977i\)
\(L(\frac12)\) \(\approx\) \(0.520100 + 0.736977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.203i)T \)
103 \( 1 + (-0.0732 + 10.1i)T \)
good2 \( 1 + (-1.38 + 0.587i)T + (1.39 - 1.43i)T^{2} \)
5 \( 1 + (-0.430 + 0.304i)T + (1.66 - 4.71i)T^{2} \)
7 \( 1 + (0.995 - 0.385i)T + (5.17 - 4.71i)T^{2} \)
11 \( 1 + (-0.481 + 0.637i)T + (-3.01 - 10.5i)T^{2} \)
13 \( 1 + (-0.00532 + 0.0342i)T + (-12.3 - 3.94i)T^{2} \)
17 \( 1 + (5.38 + 1.89i)T + (13.2 + 10.6i)T^{2} \)
19 \( 1 + (1.71 - 0.105i)T + (18.8 - 2.33i)T^{2} \)
23 \( 1 + (-0.730 - 5.89i)T + (-22.3 + 5.61i)T^{2} \)
29 \( 1 + (3.34 - 7.26i)T + (-18.8 - 22.0i)T^{2} \)
31 \( 1 + (8.08 - 0.249i)T + (30.9 - 1.90i)T^{2} \)
37 \( 1 + (-7.56 - 8.29i)T + (-3.41 + 36.8i)T^{2} \)
41 \( 1 + (-5.72 - 0.530i)T + (40.3 + 7.53i)T^{2} \)
43 \( 1 + (3.16 + 9.95i)T + (-35.0 + 24.8i)T^{2} \)
47 \( 1 + 0.588T + 47T^{2} \)
53 \( 1 + (-2.34 - 3.53i)T + (-20.6 + 48.8i)T^{2} \)
59 \( 1 + (1.49 - 3.86i)T + (-43.6 - 39.7i)T^{2} \)
61 \( 1 + (2.06 + 2.40i)T + (-9.35 + 60.2i)T^{2} \)
67 \( 1 + (-5.65 + 0.878i)T + (63.8 - 20.3i)T^{2} \)
71 \( 1 + (9.26 + 6.55i)T + (23.5 + 66.9i)T^{2} \)
73 \( 1 + (-4.60 - 0.426i)T + (71.7 + 13.4i)T^{2} \)
79 \( 1 + (-7.93 + 5.61i)T + (26.2 - 74.5i)T^{2} \)
83 \( 1 + (-4.07 - 0.632i)T + (79.0 + 25.1i)T^{2} \)
89 \( 1 + (3.74 - 13.1i)T + (-75.6 - 46.8i)T^{2} \)
97 \( 1 + (-5.11 + 5.96i)T + (-14.8 - 95.8i)T^{2} \)
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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

−10.78405989532811422649938667471, −9.452099046995239775757234610227, −8.981529639229848529761674761439, −7.56900281866381283972488117015, −6.61040059884340244613034816275, −5.69016200737021300799042213044, −5.08922750668556758476444873167, −4.15020964651250259656552175086, −3.25057565449875982703935369296, −1.78739360158287390284436484482, 0.33617254009777418429987255516, 2.25397152543750838762338622317, 4.02509448916161702166228355240, 4.44221203603907120667757924452, 5.56489896899572427003331949852, 6.33824090515626138362628473634, 6.69231483786540576223268007729, 7.81328230516798332338226791364, 9.190083567647988636375027851689, 9.903504668884538306949378545958

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