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Properties

Label 2-42e2-7.2-c1-0-15
Degree $2$
Conductor $1764$
Sign $-0.605 + 0.795i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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 + (1 + 1.73i)11-s − 6·13-s + (−2 − 3.46i)17-s + (2 − 3.46i)19-s + (1 − 1.73i)23-s + (−5.49 − 9.52i)25-s + 2·29-s + (−1 + 1.73i)37-s − 4·43-s + (6 − 10.3i)47-s + (−3 − 5.19i)53-s + 7.99·55-s + (−4 − 6.92i)59-s + (−3 + 5.19i)61-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s + (0.301 + 0.522i)11-s − 1.66·13-s + (−0.485 − 0.840i)17-s + (0.458 − 0.794i)19-s + (0.208 − 0.361i)23-s + (−1.09 − 1.90i)25-s + 0.371·29-s + (−0.164 + 0.284i)37-s − 0.609·43-s + (0.875 − 1.51i)47-s + (−0.412 − 0.713i)53-s + 1.07·55-s + (−0.520 − 0.901i)59-s + (−0.384 + 0.665i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.486002552\)
\(L(\frac12)\) \(\approx\) \(1.486002552\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.088866418172523078742275363020, −8.474960798267070556137058409536, −7.36180216667961645816901241903, −6.70936863906001212550080772473, −5.48859671670230889333768251571, −4.92245345594503368454737758928, −4.40863314865487492874741955917, −2.72475699143067748154392115129, −1.83407674202403525348031921949, −0.52077666071310926347477479911, 1.74653402820437624488130484024, 2.69322888446809988694895972122, 3.44113408142490314830604011824, 4.68669133718757697812104937707, 5.83702378265161556127313160312, 6.27747580983298863161239921394, 7.22354982346805608354521183027, 7.72554273198192039092556032398, 8.991926932846871348188833498289, 9.696858559405354909242286375666

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