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Properties

Label 2-921-921.5-c1-0-6
Degree $2$
Conductor $921$
Sign $-0.276 - 0.960i$
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  + (−1.16 − 1.28i)3-s + (−0.664 + 1.88i)4-s + (−1.93 − 4.85i)7-s + (−0.276 + 2.98i)9-s + (3.19 − 1.35i)12-s + (0.734 − 0.444i)13-s + (−3.11 − 2.50i)16-s + (−0.986 + 6.35i)19-s + (−3.95 + 8.14i)21-s + (4.99 + 0.102i)25-s + (4.14 − 3.13i)27-s + (10.4 − 0.428i)28-s + (−7.50 + 1.00i)31-s + (−5.45 − 2.50i)36-s + (−3.47 + 10.5i)37-s + ⋯
L(s)  = 1  + (−0.673 − 0.739i)3-s + (−0.332 + 0.943i)4-s + (−0.732 − 1.83i)7-s + (−0.0922 + 0.995i)9-s + (0.920 − 0.389i)12-s + (0.203 − 0.123i)13-s + (−0.779 − 0.626i)16-s + (−0.226 + 1.45i)19-s + (−0.861 + 1.77i)21-s + (0.999 + 0.0205i)25-s + (0.798 − 0.602i)27-s + (1.97 − 0.0810i)28-s + (−1.34 + 0.180i)31-s + (−0.908 − 0.417i)36-s + (−0.571 + 1.73i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.276 - 0.960i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ -0.276 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211581 + 0.281135i\)
\(L(\frac12)\) \(\approx\) \(0.211581 + 0.281135i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 + 1.28i)T \)
307 \( 1 + (13.5 - 11.1i)T \)
good2 \( 1 + (0.664 - 1.88i)T^{2} \)
5 \( 1 + (-4.99 - 0.102i)T^{2} \)
7 \( 1 + (1.93 + 4.85i)T + (-5.07 + 4.82i)T^{2} \)
11 \( 1 + (-10.9 - 0.902i)T^{2} \)
13 \( 1 + (-0.734 + 0.444i)T + (6.03 - 11.5i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.986 - 6.35i)T + (-18.1 - 5.75i)T^{2} \)
23 \( 1 + (19.7 + 11.7i)T^{2} \)
29 \( 1 + (-18.4 + 22.4i)T^{2} \)
31 \( 1 + (7.50 - 1.00i)T + (29.9 - 8.17i)T^{2} \)
37 \( 1 + (3.47 - 10.5i)T + (-29.7 - 21.9i)T^{2} \)
41 \( 1 + (4.62 + 40.7i)T^{2} \)
43 \( 1 + (4.42 - 6.53i)T + (-15.9 - 39.9i)T^{2} \)
47 \( 1 + (-43.4 + 17.8i)T^{2} \)
53 \( 1 + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (40.1 - 43.1i)T^{2} \)
61 \( 1 + (11.6 - 8.40i)T + (19.0 - 57.9i)T^{2} \)
67 \( 1 + (1.88 - 0.0966i)T + (66.6 - 6.86i)T^{2} \)
71 \( 1 + (70.2 - 10.1i)T^{2} \)
73 \( 1 + (-13.6 - 7.52i)T + (39.0 + 61.6i)T^{2} \)
79 \( 1 + (-13.0 + 3.28i)T + (69.5 - 37.3i)T^{2} \)
83 \( 1 + (66.7 - 49.3i)T^{2} \)
89 \( 1 + (-52.1 - 72.1i)T^{2} \)
97 \( 1 + (-2.66 - 17.1i)T + (-92.4 + 29.4i)T^{2} \)
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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

−10.48912156336342373010896442029, −9.616000440891691041864819976913, −8.328809474873712920871198596645, −7.69440154317043240406514039240, −6.96589018197790517306962371629, −6.34165960579315473982061342973, −4.97051255409955334468934216881, −3.98486374794244105605298074034, −3.14861679830717984661498658635, −1.29951909718392378817808218026, 0.19466802482957484139063185248, 2.19672852481211863417335599242, 3.48991541261474949834008928463, 4.86760889133796612341223016378, 5.40085450326981691057930494728, 6.15462296314827257033711723454, 6.86353034222557595730588879044, 8.776792516173450389242560178560, 9.096600592834316775189258881536, 9.662917004038960455516228942550

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