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Properties

Label 2-30e2-5.3-c2-0-8
Degree $2$
Conductor $900$
Sign $0.793 + 0.608i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.67 + 3.67i)7-s − 6·11-s + (−6.12 − 6.12i)13-s + (22.0 − 22.0i)17-s + 25i·19-s + (−7.34 − 7.34i)23-s − 42i·29-s + 49·31-s + (−4.89 + 4.89i)37-s + 60·41-s + (1.22 + 1.22i)43-s + (51.4 − 51.4i)47-s + 22i·49-s + (−14.6 − 14.6i)53-s + 78i·59-s + ⋯
L(s)  = 1  + (−0.524 + 0.524i)7-s − 0.545·11-s + (−0.471 − 0.471i)13-s + (1.29 − 1.29i)17-s + 1.31i·19-s + (−0.319 − 0.319i)23-s − 1.44i·29-s + 1.58·31-s + (−0.132 + 0.132i)37-s + 1.46·41-s + (0.0284 + 0.0284i)43-s + (1.09 − 1.09i)47-s + 0.448i·49-s + (−0.277 − 0.277i)53-s + 1.32i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.793 + 0.608i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.531940735\)
\(L(\frac12)\) \(\approx\) \(1.531940735\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (3.67 - 3.67i)T - 49iT^{2} \)
11 \( 1 + 6T + 121T^{2} \)
13 \( 1 + (6.12 + 6.12i)T + 169iT^{2} \)
17 \( 1 + (-22.0 + 22.0i)T - 289iT^{2} \)
19 \( 1 - 25iT - 361T^{2} \)
23 \( 1 + (7.34 + 7.34i)T + 529iT^{2} \)
29 \( 1 + 42iT - 841T^{2} \)
31 \( 1 - 49T + 961T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 1.36e3iT^{2} \)
41 \( 1 - 60T + 1.68e3T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + 1.84e3iT^{2} \)
47 \( 1 + (-51.4 + 51.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (14.6 + 14.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 78iT - 3.48e3T^{2} \)
61 \( 1 + 13T + 3.72e3T^{2} \)
67 \( 1 + (-52.6 + 52.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 60T + 5.04e3T^{2} \)
73 \( 1 + (63.6 + 63.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 106iT - 6.24e3T^{2} \)
83 \( 1 + (80.8 + 80.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 60iT - 7.92e3T^{2} \)
97 \( 1 + (-121. + 121. i)T - 9.40e3iT^{2} \)
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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

−10.00031080568071623578550029439, −9.088290058119766811528810005162, −7.958259147265308122406966368601, −7.51701237004855124715625481107, −6.16833970269333894056868881908, −5.59336292889304353720542595730, −4.51881192917523436684770222561, −3.22141394689632196196065859934, −2.39944491838463665816864519174, −0.62433609540893692451041309956, 1.00066329904247151785367325454, 2.55384989819853198587686994199, 3.61222757262658228598509004210, 4.64315536533644176175737550283, 5.66466411575277738572812528503, 6.63827064918375628409919743023, 7.43380719945086910571463703267, 8.263813705132773388755955085367, 9.267626172217900042743540108206, 10.05170284162128063459486566172

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