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Properties

Label 2-927-103.49-c1-0-38
Degree $2$
Conductor $927$
Sign $-0.214 + 0.976i$
Analytic cond. $7.40213$
Root an. cond. $2.72068$
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  + (1.90 − 0.606i)4-s + (−2.53 − 3.83i)7-s + (0.0866 − 0.174i)13-s + (3.26 − 2.31i)16-s + (−2.00 − 2.34i)19-s + (−4.40 − 2.36i)25-s + (−7.16 − 5.76i)28-s + (−0.248 − 2.68i)31-s + (−5.75 − 7.62i)37-s + (11.8 + 1.47i)43-s + (−5.50 + 13.0i)49-s + (0.0596 − 0.384i)52-s + (−6.96 + 4.31i)61-s + (4.82 − 6.38i)64-s + (13.6 + 0.841i)67-s + ⋯
L(s)  = 1  + (0.952 − 0.303i)4-s + (−0.959 − 1.44i)7-s + (0.0240 − 0.0482i)13-s + (0.816 − 0.577i)16-s + (−0.460 − 0.536i)19-s + (−0.881 − 0.473i)25-s + (−1.35 − 1.08i)28-s + (−0.0446 − 0.481i)31-s + (−0.946 − 1.25i)37-s + (1.81 + 0.224i)43-s + (−0.786 + 1.85i)49-s + (0.00827 − 0.0532i)52-s + (−0.891 + 0.552i)61-s + (0.602 − 0.798i)64-s + (1.66 + 0.102i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.214 + 0.976i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.214 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939533 - 1.16849i\)
\(L(\frac12)\) \(\approx\) \(0.939533 - 1.16849i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 + (-10.1 - 0.490i)T \)
good2 \( 1 + (-1.90 + 0.606i)T^{2} \)
5 \( 1 + (4.40 + 2.36i)T^{2} \)
7 \( 1 + (2.53 + 3.83i)T + (-2.72 + 6.44i)T^{2} \)
11 \( 1 + (2.35 - 10.7i)T^{2} \)
13 \( 1 + (-0.0866 + 0.174i)T + (-7.83 - 10.3i)T^{2} \)
17 \( 1 + (-16.9 + 1.04i)T^{2} \)
19 \( 1 + (2.00 + 2.34i)T + (-2.91 + 18.7i)T^{2} \)
23 \( 1 + (16.9 - 15.4i)T^{2} \)
29 \( 1 + (-0.893 - 28.9i)T^{2} \)
31 \( 1 + (0.248 + 2.68i)T + (-30.4 + 5.69i)T^{2} \)
37 \( 1 + (5.75 + 7.62i)T + (-10.1 + 35.5i)T^{2} \)
41 \( 1 + (36.1 - 19.3i)T^{2} \)
43 \( 1 + (-11.8 - 1.47i)T + (41.7 + 10.4i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-41.2 - 33.2i)T^{2} \)
59 \( 1 + (-22.9 - 54.3i)T^{2} \)
61 \( 1 + (6.96 - 4.31i)T + (27.1 - 54.6i)T^{2} \)
67 \( 1 + (-13.6 - 0.841i)T + (66.4 + 8.23i)T^{2} \)
71 \( 1 + (-2.18 + 70.9i)T^{2} \)
73 \( 1 + (-4.42 + 15.5i)T + (-62.0 - 38.4i)T^{2} \)
79 \( 1 + (-4.55 - 16.0i)T + (-67.1 + 41.5i)T^{2} \)
83 \( 1 + (82.3 - 10.1i)T^{2} \)
89 \( 1 + (8.21 - 88.6i)T^{2} \)
97 \( 1 + (-0.591 + 19.1i)T + (-96.8 - 5.97i)T^{2} \)
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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

−9.999714131505760635056201784628, −9.222856834997826501859190826215, −7.85466491954168373754985955865, −7.20079676179749625365885857251, −6.51598426104421324481640255272, −5.72206430826665008592781600301, −4.30480185791205369769865398507, −3.42033717092342976355521860767, −2.21817599932752162529288126426, −0.65954117042426737392659079468, 1.91492384007078110032345774296, 2.83894261626523623250544582614, 3.73331180060024612758092496357, 5.34433003019598094604613488506, 6.11354184341072425892520005786, 6.73669109206618636918597041749, 7.80343298952876460829004207660, 8.642922614105578812581414502523, 9.459382471933058881261777760837, 10.29920381262394821778248056520

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