Properties

Label 2-1216-19.11-c1-0-29
Degree $2$
Conductor $1216$
Sign $0.0977 + 0.995i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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)3-s + (1.5 − 2.59i)5-s + (1 + 1.73i)9-s − 4·11-s + (−2.5 − 4.33i)13-s + (1.5 + 2.59i)15-s + (2.5 − 4.33i)17-s + (4 − 1.73i)19-s + (−0.5 − 0.866i)23-s + (−2 − 3.46i)25-s − 5·27-s + (1.5 + 2.59i)29-s − 4·31-s + (2 − 3.46i)33-s − 2·37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 − 1.16i)5-s + (0.333 + 0.577i)9-s − 1.20·11-s + (−0.693 − 1.20i)13-s + (0.387 + 0.670i)15-s + (0.606 − 1.05i)17-s + (0.917 − 0.397i)19-s + (−0.104 − 0.180i)23-s + (−0.400 − 0.692i)25-s − 0.962·27-s + (0.278 + 0.482i)29-s − 0.718·31-s + (0.348 − 0.603i)33-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.0977 + 0.995i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.288304176\)
\(L(\frac12)\) \(\approx\) \(1.288304176\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4 + 1.73i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.5 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.640916440371295828170286907160, −8.895873358642018484383126156128, −7.80740443640755218767881545153, −7.35394018323993776704115904559, −5.71773145201637802273942533246, −5.01657411371967860326931911384, −4.98646779021239085606869713691, −3.28109446348280584968007258376, −2.12525329649672895441203633233, −0.56279527096814167375525697871, 1.56284473853429568989784086135, 2.60821114431416979771409002139, 3.63621842988232333853591875633, 4.94281694520370834959707764673, 6.06109566245173621023143233127, 6.46276062270330181950929496883, 7.44493612495742036205885590942, 7.950334559990657000152670075033, 9.580556452842616133874707824549, 9.715180820042853961624359963375

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