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Properties

Label 2-921-921.719-c0-0-0
Degree $2$
Conductor $921$
Sign $0.992 - 0.122i$
Analytic cond. $0.459638$
Root an. cond. $0.677966$
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  + (−0.273 − 0.961i)3-s + (0.445 + 0.895i)4-s + (1.93 + 0.361i)7-s + (−0.850 + 0.526i)9-s + (0.739 − 0.673i)12-s + (−0.890 + 1.17i)13-s + (−0.602 + 0.798i)16-s + (−0.111 + 0.147i)19-s + (−0.181 − 1.95i)21-s + (−0.602 − 0.798i)25-s + (0.739 + 0.673i)27-s + (0.538 + 1.89i)28-s + (0.538 − 1.89i)31-s + (−0.850 − 0.526i)36-s + (−0.876 − 0.163i)37-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)3-s + (0.445 + 0.895i)4-s + (1.93 + 0.361i)7-s + (−0.850 + 0.526i)9-s + (0.739 − 0.673i)12-s + (−0.890 + 1.17i)13-s + (−0.602 + 0.798i)16-s + (−0.111 + 0.147i)19-s + (−0.181 − 1.95i)21-s + (−0.602 − 0.798i)25-s + (0.739 + 0.673i)27-s + (0.538 + 1.89i)28-s + (0.538 − 1.89i)31-s + (−0.850 − 0.526i)36-s + (−0.876 − 0.163i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(0.459638\)
Root analytic conductor: \(0.677966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :0),\ 0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134631205\)
\(L(\frac12)\) \(\approx\) \(1.134631205\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.273 + 0.961i)T \)
307 \( 1 + (0.273 + 0.961i)T \)
good2 \( 1 + (-0.445 - 0.895i)T^{2} \)
5 \( 1 + (0.602 + 0.798i)T^{2} \)
7 \( 1 + (-1.93 - 0.361i)T + (0.932 + 0.361i)T^{2} \)
11 \( 1 + (0.850 + 0.526i)T^{2} \)
13 \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
23 \( 1 + (-0.445 + 0.895i)T^{2} \)
29 \( 1 + (-0.445 - 0.895i)T^{2} \)
31 \( 1 + (-0.538 + 1.89i)T + (-0.850 - 0.526i)T^{2} \)
37 \( 1 + (0.876 + 0.163i)T + (0.932 + 0.361i)T^{2} \)
41 \( 1 + (-0.932 + 0.361i)T^{2} \)
43 \( 1 + (-0.0822 + 0.887i)T + (-0.982 - 0.183i)T^{2} \)
47 \( 1 + (0.273 + 0.961i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.739 - 0.673i)T^{2} \)
61 \( 1 + (-0.136 + 1.47i)T + (-0.982 - 0.183i)T^{2} \)
67 \( 1 + (0.890 - 0.811i)T + (0.0922 - 0.995i)T^{2} \)
71 \( 1 + (0.982 - 0.183i)T^{2} \)
73 \( 1 + (-1.18 + 1.56i)T + (-0.273 - 0.961i)T^{2} \)
79 \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \)
83 \( 1 + (-0.932 + 0.361i)T^{2} \)
89 \( 1 + (-0.0922 + 0.995i)T^{2} \)
97 \( 1 + (0.890 + 1.17i)T + (-0.273 + 0.961i)T^{2} \)
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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

−10.66013061222980905348963556155, −9.174193365004256194217258397019, −8.264716786341896455568851773472, −7.81272892227086912890035156695, −7.10970173896805090800275462196, −6.14036667988100407482118444993, −5.02291263797050266059314444937, −4.14459641378721205374496957247, −2.35525142165733291762264592423, −1.89163828296291953148190556310, 1.37462592640506884605223682307, 2.80350446759414440929234350240, 4.30443988720521313571974550093, 5.23209741233204234477316974998, 5.40163631825174010229567169902, 6.84264252459581765320725765814, 7.81285459017062757723301640898, 8.629976108606625614687314941397, 9.693134108257429379871408292788, 10.48120277054271750179981209380

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