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Properties

Label 2-30e2-45.34-c1-0-9
Degree $2$
Conductor $900$
Sign $0.940 + 0.340i$
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  + (−0.0977 + 1.72i)3-s + (−2.94 − 1.70i)7-s + (−2.98 − 0.338i)9-s + (2.20 − 3.81i)11-s + (1.78 − 1.03i)13-s + 1.40i·17-s + 6.35·19-s + (3.22 − 4.92i)21-s + (0.0933 − 0.0539i)23-s + (0.875 − 5.12i)27-s + (4.54 − 7.86i)29-s + (−1.53 − 2.65i)31-s + (6.37 + 4.17i)33-s − 1.95i·37-s + (1.60 + 3.19i)39-s + ⋯
L(s)  = 1  + (−0.0564 + 0.998i)3-s + (−1.11 − 0.642i)7-s + (−0.993 − 0.112i)9-s + (0.663 − 1.14i)11-s + (0.495 − 0.286i)13-s + 0.339i·17-s + 1.45·19-s + (0.704 − 1.07i)21-s + (0.0194 − 0.0112i)23-s + (0.168 − 0.985i)27-s + (0.843 − 1.46i)29-s + (−0.275 − 0.476i)31-s + (1.11 + 0.727i)33-s − 0.321i·37-s + (0.257 + 0.510i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.940 + 0.340i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.940 + 0.340i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25774 - 0.220697i\)
\(L(\frac12)\) \(\approx\) \(1.25774 - 0.220697i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0977 - 1.72i)T \)
5 \( 1 \)
good7 \( 1 + (2.94 + 1.70i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.20 + 3.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.78 + 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.40iT - 17T^{2} \)
19 \( 1 - 6.35T + 19T^{2} \)
23 \( 1 + (-0.0933 + 0.0539i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.54 + 7.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.53 + 2.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.95iT - 37T^{2} \)
41 \( 1 + (-4.34 - 7.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.17 - 3.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.48 + 3.74i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + (6.58 + 11.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.862 + 1.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.7 + 6.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.50T + 71T^{2} \)
73 \( 1 + 5.42iT - 73T^{2} \)
79 \( 1 + (-4.71 + 8.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.77 + 4.48i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.01T + 89T^{2} \)
97 \( 1 + (-1.88 - 1.08i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.845333937200056016991557285743, −9.526179403005112122758976941587, −8.542935192269863524601639127667, −7.63009232627478760031062727720, −6.24559191287801877716014458426, −5.93674950177302022480627714241, −4.55739364399649014871684717943, −3.58902333565380660153677966615, −3.04513183317094257718618145871, −0.70345028563848291900607899156, 1.27720682229524392100115710639, 2.58101469357121646244081682106, 3.55840364548148056804794592616, 5.07495202992091393405005345840, 5.98826513535238684912220051902, 6.89568501937246602901870890751, 7.27793425682415263310405200330, 8.589468851797260492016947505830, 9.238846555270095680961705721734, 9.974364499841239960390772314904

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