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Properties

Label 2-30e2-45.4-c1-0-16
Degree $2$
Conductor $900$
Sign $0.132 + 0.991i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (1.28 − 1.16i)3-s + (4.32 − 2.49i)7-s + (0.304 − 2.98i)9-s + (−1.99 − 3.46i)11-s + (−1.33 − 0.771i)13-s + 6.99i·17-s + 2.25·19-s + (2.66 − 8.23i)21-s + (−6.75 − 3.89i)23-s + (−3.07 − 4.18i)27-s + (3.08 + 5.33i)29-s + (0.271 − 0.470i)31-s + (−6.58 − 2.12i)33-s + 6.25i·37-s + (−2.61 + 0.559i)39-s + ⋯
L(s)  = 1  + (0.742 − 0.670i)3-s + (1.63 − 0.944i)7-s + (0.101 − 0.994i)9-s + (−0.602 − 1.04i)11-s + (−0.370 − 0.213i)13-s + 1.69i·17-s + 0.517·19-s + (0.580 − 1.79i)21-s + (−1.40 − 0.812i)23-s + (−0.591 − 0.806i)27-s + (0.572 + 0.991i)29-s + (0.0487 − 0.0844i)31-s + (−1.14 − 0.370i)33-s + 1.02i·37-s + (−0.418 + 0.0896i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68363 - 1.47353i\)
\(L(\frac12)\) \(\approx\) \(1.68363 - 1.47353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 + 1.16i)T \)
5 \( 1 \)
good7 \( 1 + (-4.32 + 2.49i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.99 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.33 + 0.771i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.99iT - 17T^{2} \)
19 \( 1 - 2.25T + 19T^{2} \)
23 \( 1 + (6.75 + 3.89i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.08 - 5.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.271 + 0.470i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.25iT - 37T^{2} \)
41 \( 1 + (0.0979 - 0.169i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0747 - 0.0431i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.31 + 1.91i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.19iT - 53T^{2} \)
59 \( 1 + (-3.51 + 6.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.45 - 2.51i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.76 - 4.48i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.79T + 71T^{2} \)
73 \( 1 + 2.28iT - 73T^{2} \)
79 \( 1 + (6.32 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.0 - 6.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (8.08 - 4.66i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.11307958147032995989600663682, −8.597512015572784507793757512863, −8.212688174680221936634720220516, −7.68401850931164064822422393601, −6.63613436101264317709867094349, −5.58331963317582469906709928090, −4.41946051153876835870592374784, −3.47835527099861608884976430479, −2.12915434661188791239559860430, −1.05234382383930921761089820464, 1.99075279200829131455016158156, 2.61354531427418862891290909744, 4.19712090992363170423710233897, 4.96683210285627183737405285880, 5.52920885765272652955539689825, 7.34521783946281445637113371641, 7.80915830298589004624071657707, 8.628805260682423844005890565313, 9.545491791539879588316857236933, 10.00993394571709412123235164111

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