Properties

Label 2-2352-7.2-c1-0-2
Degree $2$
Conductor $2352$
Sign $-0.701 - 0.712i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−1 + 1.73i)5-s + (−0.499 + 0.866i)9-s + (2 + 3.46i)11-s + 2·13-s + 1.99·15-s + (−3 − 5.19i)17-s + (−2 + 3.46i)19-s + (0.500 + 0.866i)25-s + 0.999·27-s − 2·29-s + (1.99 − 3.46i)33-s + (−3 + 5.19i)37-s + (−1 − 1.73i)39-s − 2·41-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.166 + 0.288i)9-s + (0.603 + 1.04i)11-s + 0.554·13-s + 0.516·15-s + (−0.727 − 1.26i)17-s + (−0.458 + 0.794i)19-s + (0.100 + 0.173i)25-s + 0.192·27-s − 0.371·29-s + (0.348 − 0.603i)33-s + (−0.493 + 0.854i)37-s + (−0.160 − 0.277i)39-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6820223858\)
\(L(\frac12)\) \(\approx\) \(0.6820223858\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
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 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.300118321211605063923040790306, −8.385784379800755807502046996465, −7.56402726499231599570365145860, −6.87045450635048356605933690787, −6.49359849011394730112873966285, −5.37447014127236160357829629877, −4.44575502710521987012213463275, −3.55719680809943734518882248158, −2.50372792709442736496260014304, −1.44706056803683941569737165584, 0.24886185022786222439582601598, 1.51572481975669967895345570004, 3.03725136147849304515119055929, 4.08793253634363500271815482629, 4.43352542860517706390067571029, 5.64810322739201402753811284843, 6.16181959119288311607727680876, 7.09330532901617981437603416994, 8.256156235302716107394549011237, 8.752076864244550689075817986574

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