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Properties

Label 2-921-921.137-c1-0-13
Degree $2$
Conductor $921$
Sign $-0.328 - 0.944i$
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  + (−0.625 − 1.61i)3-s + (1.81 + 0.835i)4-s + (−2.22 + 0.229i)7-s + (−2.21 + 2.02i)9-s + (0.213 − 3.45i)12-s + (−6.35 + 2.38i)13-s + (2.60 + 3.03i)16-s + (−2.84 + 8.08i)19-s + (1.76 + 3.45i)21-s + (−4.93 + 0.817i)25-s + (4.65 + 2.31i)27-s + (−4.23 − 1.44i)28-s + (8.69 − 4.78i)31-s + (−5.71 + 1.81i)36-s + (−4.94 − 3.35i)37-s + ⋯
L(s)  = 1  + (−0.361 − 0.932i)3-s + (0.908 + 0.417i)4-s + (−0.841 + 0.0866i)7-s + (−0.739 + 0.673i)9-s + (0.0615 − 0.998i)12-s + (−1.76 + 0.662i)13-s + (0.650 + 0.759i)16-s + (−0.653 + 1.85i)19-s + (0.384 + 0.753i)21-s + (−0.986 + 0.163i)25-s + (0.895 + 0.445i)27-s + (−0.800 − 0.272i)28-s + (1.56 − 0.859i)31-s + (−0.952 + 0.303i)36-s + (−0.813 − 0.551i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ -0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368215 + 0.517732i\)
\(L(\frac12)\) \(\approx\) \(0.368215 + 0.517732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.625 + 1.61i)T \)
307 \( 1 + (16.6 - 5.51i)T \)
good2 \( 1 + (-1.81 - 0.835i)T^{2} \)
5 \( 1 + (4.93 - 0.817i)T^{2} \)
7 \( 1 + (2.22 - 0.229i)T + (6.85 - 1.42i)T^{2} \)
11 \( 1 + (-8.70 + 6.71i)T^{2} \)
13 \( 1 + (6.35 - 2.38i)T + (9.78 - 8.55i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.84 - 8.08i)T + (-14.8 - 11.9i)T^{2} \)
23 \( 1 + (-9.82 + 20.7i)T^{2} \)
29 \( 1 + (20.6 + 20.4i)T^{2} \)
31 \( 1 + (-8.69 + 4.78i)T + (16.5 - 26.1i)T^{2} \)
37 \( 1 + (4.94 + 3.35i)T + (13.7 + 34.3i)T^{2} \)
41 \( 1 + (-25.3 - 32.2i)T^{2} \)
43 \( 1 + (9.68 - 0.497i)T + (42.7 - 4.40i)T^{2} \)
47 \( 1 + (-46.9 + 0.964i)T^{2} \)
53 \( 1 + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (56.5 + 16.7i)T^{2} \)
61 \( 1 + (-11.4 - 3.50i)T + (50.5 + 34.2i)T^{2} \)
67 \( 1 + (-1.93 + 4.43i)T + (-45.6 - 49.0i)T^{2} \)
71 \( 1 + (-29.0 - 64.8i)T^{2} \)
73 \( 1 + (-8.47 - 6.96i)T + (14.1 + 71.6i)T^{2} \)
79 \( 1 + (3.27 - 7.72i)T + (-54.9 - 56.7i)T^{2} \)
83 \( 1 + (-30.7 + 77.0i)T^{2} \)
89 \( 1 + (-26.1 + 85.0i)T^{2} \)
97 \( 1 + (-0.201 - 0.572i)T + (-75.5 + 60.8i)T^{2} \)
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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

−10.27035672421144647027380252339, −9.726863656232074446606196734356, −8.291824605504767285367276892770, −7.71582189101569761057397317400, −6.81883228492769865120771253093, −6.33872269980115739102515310398, −5.37697080197806756363762396049, −3.89018776565702668426771247616, −2.62899502652283854330543377162, −1.85146852742587179363898885498, 0.27545053641311345183642444791, 2.48563279589513937670939979715, 3.23020019541403596484935073468, 4.70561722203200915783961805021, 5.32710734177077210179014419985, 6.51000629640533828378502864770, 6.92010681346693263179225379532, 8.182935352721247697293081601337, 9.343899254242111844487112702522, 10.07180666398959891948449219717

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