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Properties

Label 2-921-921.14-c1-0-6
Degree $2$
Conductor $921$
Sign $0.695 - 0.718i$
Analytic cond. $7.35422$
Root an. cond. $2.71186$
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  + (−0.911 − 1.47i)3-s + (−1.98 − 0.245i)4-s + (−3.74 − 3.27i)7-s + (−1.33 + 2.68i)9-s + (1.44 + 3.14i)12-s + (−4.42 + 3.19i)13-s + (3.87 + 0.975i)16-s + (2.35 − 2.42i)19-s + (−1.41 + 8.51i)21-s + (−2.04 − 4.56i)25-s + (5.17 − 0.479i)27-s + (6.63 + 7.43i)28-s + (4.81 + 1.92i)31-s + (3.31 − 5.00i)36-s + (0.257 − 0.425i)37-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)3-s + (−0.992 − 0.122i)4-s + (−1.41 − 1.23i)7-s + (−0.445 + 0.895i)9-s + (0.417 + 0.908i)12-s + (−1.22 + 0.887i)13-s + (0.969 + 0.243i)16-s + (0.540 − 0.556i)19-s + (−0.307 + 1.85i)21-s + (−0.408 − 0.912i)25-s + (0.995 − 0.0922i)27-s + (1.25 + 1.40i)28-s + (0.865 + 0.345i)31-s + (0.552 − 0.833i)36-s + (0.0423 − 0.0699i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256233 + 0.108549i\)
\(L(\frac12)\) \(\approx\) \(0.256233 + 0.108549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.911 + 1.47i)T \)
307 \( 1 + (-14.4 + 9.94i)T \)
good2 \( 1 + (1.98 + 0.245i)T^{2} \)
5 \( 1 + (2.04 + 4.56i)T^{2} \)
7 \( 1 + (3.74 + 3.27i)T + (0.931 + 6.93i)T^{2} \)
11 \( 1 + (1.23 + 10.9i)T^{2} \)
13 \( 1 + (4.42 - 3.19i)T + (4.06 - 12.3i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.35 + 2.42i)T + (-0.585 - 18.9i)T^{2} \)
23 \( 1 + (1.18 - 22.9i)T^{2} \)
29 \( 1 + (19.7 + 21.2i)T^{2} \)
31 \( 1 + (-4.81 - 1.92i)T + (22.4 + 21.3i)T^{2} \)
37 \( 1 + (-0.257 + 0.425i)T + (-17.1 - 32.7i)T^{2} \)
41 \( 1 + (-40.9 + 1.68i)T^{2} \)
43 \( 1 + (-4.09 - 1.53i)T + (32.3 + 28.3i)T^{2} \)
47 \( 1 + (-46.5 - 6.73i)T^{2} \)
53 \( 1 + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-25.2 - 53.3i)T^{2} \)
61 \( 1 + (1.46 + 2.60i)T + (-31.5 + 52.1i)T^{2} \)
67 \( 1 + (1.02 - 3.73i)T + (-57.6 - 34.0i)T^{2} \)
71 \( 1 + (13.7 + 69.6i)T^{2} \)
73 \( 1 + (-1.58 - 15.3i)T + (-71.4 + 14.8i)T^{2} \)
79 \( 1 + (5.88 - 16.6i)T + (-61.5 - 49.5i)T^{2} \)
83 \( 1 + (38.5 - 73.5i)T^{2} \)
89 \( 1 + (77.5 - 43.7i)T^{2} \)
97 \( 1 + (13.4 + 13.9i)T + (-2.98 + 96.9i)T^{2} \)
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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

−9.931564133319690212571734384546, −9.655154999114933407622818913964, −8.471676429879001855945452049766, −7.40746968429689983751847320452, −6.86846629006869651764415329703, −6.00412344160988853510736438291, −4.84959766749530563556080370285, −4.05266979284035329225796354755, −2.71665098925843585675095127834, −0.897485896841844167444822499767, 0.19658903358836863166137348928, 2.85550123891901168062115177549, 3.57014911591865970900587174484, 4.79678685847686937120449273374, 5.55953054895036405216239940350, 6.13142410746703256790431880899, 7.50855000689606038102049174838, 8.619955488281075773992225628413, 9.438300127274114509267836055627, 9.743846288528318297413194547079

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