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Properties

Label 2-1386-231.32-c1-0-7
Degree $2$
Conductor $1386$
Sign $-0.983 - 0.181i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.346 − 0.199i)5-s + (1.03 + 2.43i)7-s − 0.999·8-s + (0.346 + 0.199i)10-s + (−3.01 − 1.38i)11-s + 0.164i·13-s + (−1.58 + 2.11i)14-s + (−0.5 − 0.866i)16-s + (−0.906 + 1.57i)17-s + (−5.41 + 3.12i)19-s + 0.399i·20-s + (−0.308 − 3.30i)22-s + (−3.76 + 2.17i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.154 − 0.0894i)5-s + (0.392 + 0.919i)7-s − 0.353·8-s + (0.109 + 0.0632i)10-s + (−0.908 − 0.417i)11-s + 0.0456i·13-s + (−0.424 + 0.565i)14-s + (−0.125 − 0.216i)16-s + (−0.219 + 0.380i)17-s + (−1.24 + 0.717i)19-s + 0.0894i·20-s + (−0.0656 − 0.704i)22-s + (−0.785 + 0.453i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.983 - 0.181i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.983 - 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.201583028\)
\(L(\frac12)\) \(\approx\) \(1.201583028\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.03 - 2.43i)T \)
11 \( 1 + (3.01 + 1.38i)T \)
good5 \( 1 + (-0.346 + 0.199i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.164iT - 13T^{2} \)
17 \( 1 + (0.906 - 1.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.41 - 3.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.76 - 2.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + (0.141 - 0.245i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.40 - 4.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.23T + 41T^{2} \)
43 \( 1 - 2.07iT - 43T^{2} \)
47 \( 1 + (0.367 - 0.212i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.71 + 4.45i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.92 + 3.99i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.10 - 3.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0327 + 0.0567i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.43iT - 71T^{2} \)
73 \( 1 + (-7.21 - 4.16i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.531 + 0.306i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.91T + 83T^{2} \)
89 \( 1 + (8.89 - 5.13i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−9.813728062445396063516331577403, −8.963751777530303714460459223488, −8.133271321721969282311012232666, −7.78307028321484510521306337430, −6.38578423267053574455090383033, −5.89563248031016611450576422370, −5.07884021454024485964850539864, −4.16718917223480400231417146602, −2.96843907784841459332114429556, −1.88161088053484296470245137018, 0.40466886618828163386324429090, 1.99577270758391142463308766972, 2.83953056815188950428675896129, 4.29902694992762520683158635492, 4.55650619280601896709569837433, 5.79349822530829025202173658018, 6.67718684048390950718213029197, 7.63496381080746155189131145459, 8.363667630727126778192065153225, 9.415295878076548798886411891255

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