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Properties

Label 2-1512-63.59-c1-0-2
Degree $2$
Conductor $1512$
Sign $0.803 - 0.594i$
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  − 4.11·5-s + (−2.54 + 0.711i)7-s − 5.82i·11-s + (−2.52 − 1.45i)13-s + (−1.58 + 2.73i)17-s + (0.722 − 0.417i)19-s + 7.09i·23-s + 11.9·25-s + (1.91 − 1.10i)29-s + (3.66 − 2.11i)31-s + (10.4 − 2.92i)35-s + (1.82 + 3.16i)37-s + (−2.04 + 3.54i)41-s + (0.155 + 0.269i)43-s + (−0.502 + 0.870i)47-s + ⋯
L(s)  = 1  − 1.84·5-s + (−0.963 + 0.269i)7-s − 1.75i·11-s + (−0.699 − 0.403i)13-s + (−0.383 + 0.664i)17-s + (0.165 − 0.0956i)19-s + 1.47i·23-s + 2.38·25-s + (0.355 − 0.205i)29-s + (0.658 − 0.380i)31-s + (1.77 − 0.495i)35-s + (0.300 + 0.519i)37-s + (−0.319 + 0.554i)41-s + (0.0237 + 0.0410i)43-s + (−0.0732 + 0.126i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6298328013\)
\(L(\frac12)\) \(\approx\) \(0.6298328013\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 - 0.711i)T \)
good5 \( 1 + 4.11T + 5T^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (2.52 + 1.45i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.58 - 2.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.722 + 0.417i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.09iT - 23T^{2} \)
29 \( 1 + (-1.91 + 1.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.66 + 2.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.82 - 3.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.04 - 3.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.155 - 0.269i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.502 - 0.870i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.94 - 1.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.51 + 4.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.98 + 2.30i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.99 - 8.65i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (3.04 + 1.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.579 + 1.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.57 - 13.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.82 + 8.35i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.06 + 2.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.477057022051465602189442943257, −8.509435000934240735751613363122, −8.088527669172762313527084632518, −7.24214321297696121503026086377, −6.35707079118504842813309762477, −5.47643777974278768485106751681, −4.29880332341285349752829015996, −3.43115891711152425346590166872, −2.93825458003065064355409261329, −0.70562592886710885183903190308, 0.41321759876375017429246869330, 2.41168163782665534248185889767, 3.43892409471280855720834608354, 4.48293696677527593728263685536, 4.72410608784488262220549061677, 6.50224548110038454568171816369, 7.15129679846272515568318464847, 7.53024410811210022854575291018, 8.571452842135776165241939552402, 9.384431369443727308261848003958

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