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Properties

Label 2-927-103.36-c1-0-12
Degree $2$
Conductor $927$
Sign $0.173 - 0.984i$
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  + (1.55 + 1.25i)4-s + (−0.0673 + 2.18i)7-s + (−2.36 − 1.46i)13-s + (0.855 + 3.90i)16-s + (6.82 + 4.83i)19-s + (3.48 + 3.58i)25-s + (−2.84 + 3.32i)28-s + (−8.03 + 7.32i)31-s + (−5.38 − 10.8i)37-s + (1.04 + 1.57i)43-s + (2.21 + 0.136i)49-s + (−1.85 − 5.25i)52-s + (−4.10 + 14.4i)61-s + (−3.56 + 7.16i)64-s + (13.2 + 7.13i)67-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)4-s + (−0.0254 + 0.825i)7-s + (−0.656 − 0.406i)13-s + (0.213 + 0.976i)16-s + (1.56 + 1.10i)19-s + (0.696 + 0.717i)25-s + (−0.537 + 0.627i)28-s + (−1.44 + 1.31i)31-s + (−0.885 − 1.77i)37-s + (0.159 + 0.240i)43-s + (0.316 + 0.0195i)49-s + (−0.256 − 0.728i)52-s + (−0.526 + 1.84i)61-s + (−0.445 + 0.895i)64-s + (1.62 + 0.872i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36402 + 1.14428i\)
\(L(\frac12)\) \(\approx\) \(1.36402 + 1.14428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.07 + 9.29i)T \)
good2 \( 1 + (-1.55 - 1.25i)T^{2} \)
5 \( 1 + (-3.48 - 3.58i)T^{2} \)
7 \( 1 + (0.0673 - 2.18i)T + (-6.98 - 0.430i)T^{2} \)
11 \( 1 + (-1.68 + 10.8i)T^{2} \)
13 \( 1 + (2.36 + 1.46i)T + (5.79 + 11.6i)T^{2} \)
17 \( 1 + (14.9 - 8.04i)T^{2} \)
19 \( 1 + (-6.82 - 4.83i)T + (6.31 + 17.9i)T^{2} \)
23 \( 1 + (21.4 + 8.30i)T^{2} \)
29 \( 1 + (28.1 - 7.07i)T^{2} \)
31 \( 1 + (8.03 - 7.32i)T + (2.86 - 30.8i)T^{2} \)
37 \( 1 + (5.38 + 10.8i)T + (-22.2 + 29.5i)T^{2} \)
41 \( 1 + (-28.5 + 29.4i)T^{2} \)
43 \( 1 + (-1.04 - 1.57i)T + (-16.7 + 39.5i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (34.4 - 40.2i)T^{2} \)
59 \( 1 + (-58.8 + 3.63i)T^{2} \)
61 \( 1 + (4.10 - 14.4i)T + (-51.8 - 32.1i)T^{2} \)
67 \( 1 + (-13.2 - 7.13i)T + (37.0 + 55.8i)T^{2} \)
71 \( 1 + (68.8 + 17.3i)T^{2} \)
73 \( 1 + (-9.44 + 12.5i)T + (-19.9 - 70.2i)T^{2} \)
79 \( 1 + (10.5 + 13.9i)T + (-21.6 + 75.9i)T^{2} \)
83 \( 1 + (45.8 - 69.1i)T^{2} \)
89 \( 1 + (65.7 + 59.9i)T^{2} \)
97 \( 1 + (-9.20 - 2.31i)T + (85.4 + 45.8i)T^{2} \)
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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

−10.37125681071653101012664486311, −9.310324901091573703670458350840, −8.600100060386742947189580373083, −7.51489813716761903114476506076, −7.14590225127191452967050666682, −5.82300556368917113834286681168, −5.24726137006292859467729828474, −3.66395303669156452864084186435, −2.89613332696223194792536085136, −1.72791755217303241055507078839, 0.845538305457947681340484570844, 2.22966283652196398527001835115, 3.37273196202860911619817279974, 4.71096724975986489303070409757, 5.47656133715662002492393304132, 6.72940715273916627444137955816, 7.10419121127898084456731753543, 8.019210220814405536573922725947, 9.372578264670667607970090831844, 9.853855295690232130154456192801

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