Properties

Label 2-927-103.94-c0-0-0
Degree $2$
Conductor $927$
Sign $0.820 - 0.572i$
Analytic cond. $0.462633$
Root an. cond. $0.680171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.602 + 0.798i)4-s + (1.67 + 0.312i)7-s + (−1.18 − 0.221i)13-s + (−0.273 + 0.961i)16-s + (−1.25 − 0.778i)19-s + (0.0922 − 0.995i)25-s + (0.757 + 1.52i)28-s + (−0.694 − 0.197i)31-s + (0.132 − 0.342i)37-s + (0.719 + 1.85i)43-s + (1.76 + 0.683i)49-s + (−0.537 − 1.07i)52-s + (0.0505 − 0.544i)61-s + (−0.932 + 0.361i)64-s + (−0.328 − 1.75i)67-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)4-s + (1.67 + 0.312i)7-s + (−1.18 − 0.221i)13-s + (−0.273 + 0.961i)16-s + (−1.25 − 0.778i)19-s + (0.0922 − 0.995i)25-s + (0.757 + 1.52i)28-s + (−0.694 − 0.197i)31-s + (0.132 − 0.342i)37-s + (0.719 + 1.85i)43-s + (1.76 + 0.683i)49-s + (−0.537 − 1.07i)52-s + (0.0505 − 0.544i)61-s + (−0.932 + 0.361i)64-s + (−0.328 − 1.75i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.820 - 0.572i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.820 - 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.252209060\)
\(L(\frac12)\) \(\approx\) \(1.252209060\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 + (0.932 + 0.361i)T \)
good2 \( 1 + (-0.602 - 0.798i)T^{2} \)
5 \( 1 + (-0.0922 + 0.995i)T^{2} \)
7 \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \)
11 \( 1 + (0.602 + 0.798i)T^{2} \)
13 \( 1 + (1.18 + 0.221i)T + (0.932 + 0.361i)T^{2} \)
17 \( 1 + (-0.982 - 0.183i)T^{2} \)
19 \( 1 + (1.25 + 0.778i)T + (0.445 + 0.895i)T^{2} \)
23 \( 1 + (-0.602 + 0.798i)T^{2} \)
29 \( 1 + (0.0922 - 0.995i)T^{2} \)
31 \( 1 + (0.694 + 0.197i)T + (0.850 + 0.526i)T^{2} \)
37 \( 1 + (-0.132 + 0.342i)T + (-0.739 - 0.673i)T^{2} \)
41 \( 1 + (0.0922 + 0.995i)T^{2} \)
43 \( 1 + (-0.719 - 1.85i)T + (-0.739 + 0.673i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.445 - 0.895i)T^{2} \)
59 \( 1 + (0.932 - 0.361i)T^{2} \)
61 \( 1 + (-0.0505 + 0.544i)T + (-0.982 - 0.183i)T^{2} \)
67 \( 1 + (0.328 + 1.75i)T + (-0.932 + 0.361i)T^{2} \)
71 \( 1 + (-0.0922 - 0.995i)T^{2} \)
73 \( 1 + (1.29 - 1.42i)T + (-0.0922 - 0.995i)T^{2} \)
79 \( 1 + (-1.45 + 1.32i)T + (0.0922 - 0.995i)T^{2} \)
83 \( 1 + (0.932 + 0.361i)T^{2} \)
89 \( 1 + (0.273 + 0.961i)T^{2} \)
97 \( 1 + (0.0170 + 0.183i)T + (-0.982 + 0.183i)T^{2} \)
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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

−10.71737888592003911281236596825, −9.392431573090436072183303894559, −8.431182036432562980846269800470, −7.900956725948871685377543514368, −7.16577033069684558569428686970, −6.12632394453479180944003242678, −4.89750136906493161273275483498, −4.27183197950720857394253993020, −2.71232868546548135029042947807, −1.95943751010902439467531085996, 1.53259855540372491047208685002, 2.34998374434137112277526819748, 4.10803443955045825501440108235, 5.03483342721992114303534911852, 5.69877234635094862599918790326, 6.95381412919451150862332504183, 7.53217850871966068719029535620, 8.469954285755770820643698872591, 9.440423178878300930481945405894, 10.49167159626760440268691394400

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