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Properties

Label 2-30e2-100.23-c1-0-18
Degree $2$
Conductor $900$
Sign $0.299 - 0.954i$
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  + (−1.24 + 0.678i)2-s + (1.08 − 1.68i)4-s + (0.872 − 2.05i)5-s + (3.13 + 3.13i)7-s + (−0.199 + 2.82i)8-s + (0.312 + 3.14i)10-s + (3.28 + 4.52i)11-s + (0.499 + 3.15i)13-s + (−6.00 − 1.76i)14-s + (−1.66 − 3.63i)16-s + (−2.24 − 1.14i)17-s + (−0.732 − 2.25i)19-s + (−2.52 − 3.69i)20-s + (−7.14 − 3.38i)22-s + (−1.33 + 8.40i)23-s + ⋯
L(s)  = 1  + (−0.877 + 0.479i)2-s + (0.540 − 0.841i)4-s + (0.390 − 0.920i)5-s + (1.18 + 1.18i)7-s + (−0.0704 + 0.997i)8-s + (0.0989 + 0.995i)10-s + (0.990 + 1.36i)11-s + (0.138 + 0.875i)13-s + (−1.60 − 0.471i)14-s + (−0.416 − 0.909i)16-s + (−0.543 − 0.276i)17-s + (−0.167 − 0.516i)19-s + (−0.563 − 0.825i)20-s + (−1.52 − 0.721i)22-s + (−0.277 + 1.75i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01586 + 0.745606i\)
\(L(\frac12)\) \(\approx\) \(1.01586 + 0.745606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.678i)T \)
3 \( 1 \)
5 \( 1 + (-0.872 + 2.05i)T \)
good7 \( 1 + (-3.13 - 3.13i)T + 7iT^{2} \)
11 \( 1 + (-3.28 - 4.52i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.499 - 3.15i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.24 + 1.14i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.732 + 2.25i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.33 - 8.40i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (3.85 + 1.25i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.06 - 0.670i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.19 - 0.506i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.37 - 3.90i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.00 - 1.00i)T - 43iT^{2} \)
47 \( 1 + (-0.610 + 0.311i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-8.35 + 4.25i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (3.98 + 2.89i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.67 - 1.94i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.883 + 1.73i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (7.28 + 2.36i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.69 - 1.21i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.42 + 4.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-11.1 - 5.69i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (7.91 + 10.8i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.0175 - 0.0345i)T + (-57.0 + 78.4i)T^{2} \)
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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

−9.812882883574206740194034778687, −9.187256695268217082685917110904, −8.884644579769301104838150304395, −7.86793698181111876660685654690, −7.02156535473182140451937979781, −5.96666108045230119173607971021, −5.13712175240189413257142728280, −4.39817193663967386376530464927, −2.02427638744406235667860743688, −1.63404243251017512634921796712, 0.851593557924656242711610309833, 2.10264575042650625675910475127, 3.42583636379127914838459896227, 4.16921068162905114592101231191, 5.86151872960368307648299375472, 6.71702417545868882151977952286, 7.56496138515209457959836728966, 8.323685263174783947795957723157, 9.041650609354446061838310029326, 10.29868310980985941832022691546

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