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Properties

Label 2-1512-7.2-c1-0-12
Degree $2$
Conductor $1512$
Sign $-0.386 - 0.922i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−2 + 3.46i)5-s + (2 + 1.73i)7-s + (1 + 1.73i)11-s + 5·13-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (3 − 5.19i)23-s + (−5.49 − 9.52i)25-s − 6·29-s + (3.5 + 6.06i)31-s + (−10 + 3.46i)35-s + (−3.5 + 6.06i)37-s − 2·41-s − 7·43-s + (−1 + 1.73i)47-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + (0.755 + 0.654i)7-s + (0.301 + 0.522i)11-s + 1.38·13-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (0.625 − 1.08i)23-s + (−1.09 − 1.90i)25-s − 1.11·29-s + (0.628 + 1.08i)31-s + (−1.69 + 0.585i)35-s + (−0.575 + 0.996i)37-s − 0.312·41-s − 1.06·43-s + (−0.145 + 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.623867738\)
\(L(\frac12)\) \(\approx\) \(1.623867738\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15T + 97T^{2} \)
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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

−9.891103050997054822796870221026, −8.560629388506287812779049031959, −8.290977677570387414625359999282, −7.20213524718571884979162864814, −6.63604349944254916072148666766, −5.76266706380442238274247075175, −4.57937439209619076441370522601, −3.60755289181998327364524526290, −2.88706280955806689260401758043, −1.56298052508001374066927580267, 0.74239155082555296255416921322, 1.47932143978915576995787333997, 3.59989910700719067871166672192, 3.95378218285554347177572472636, 5.17272820491445669458447354842, 5.55159897711243042775569830405, 7.03371033028207807433694901342, 7.87074777864712119392797451388, 8.298330237285907210775313761393, 9.102578133149313291686741161861

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