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Properties

Label 2-30e2-15.2-c5-0-0
Degree $2$
Conductor $900$
Sign $-0.998 + 0.0618i$
Analytic cond. $144.345$
Root an. cond. $12.0143$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (104. + 104. i)7-s − 421. i·11-s + (−459. + 459. i)13-s + (−535. + 535. i)17-s − 1.81e3i·19-s + (2.93e3 + 2.93e3i)23-s + 3.14e3·29-s − 3.73e3·31-s + (−8.26e3 − 8.26e3i)37-s + 2.48e3i·41-s + (−5.54e3 + 5.54e3i)43-s + (−1.86e4 + 1.86e4i)47-s + 5.17e3i·49-s + (−5.83e3 − 5.83e3i)53-s − 3.17e4·59-s + ⋯
L(s)  = 1  + (0.808 + 0.808i)7-s − 1.05i·11-s + (−0.754 + 0.754i)13-s + (−0.449 + 0.449i)17-s − 1.15i·19-s + (1.15 + 1.15i)23-s + 0.694·29-s − 0.697·31-s + (−0.992 − 0.992i)37-s + 0.231i·41-s + (−0.457 + 0.457i)43-s + (−1.23 + 1.23i)47-s + 0.308i·49-s + (−0.285 − 0.285i)53-s − 1.18·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(144.345\)
Root analytic conductor: \(12.0143\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :5/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2607490480\)
\(L(\frac12)\) \(\approx\) \(0.2607490480\)
\(L(\frac{7}{2})\) 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 \( 1 + (-104. - 104. i)T + 1.68e4iT^{2} \)
11 \( 1 + 421. iT - 1.61e5T^{2} \)
13 \( 1 + (459. - 459. i)T - 3.71e5iT^{2} \)
17 \( 1 + (535. - 535. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.81e3iT - 2.47e6T^{2} \)
23 \( 1 + (-2.93e3 - 2.93e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 3.14e3T + 2.05e7T^{2} \)
31 \( 1 + 3.73e3T + 2.86e7T^{2} \)
37 \( 1 + (8.26e3 + 8.26e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 2.48e3iT - 1.15e8T^{2} \)
43 \( 1 + (5.54e3 - 5.54e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (1.86e4 - 1.86e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (5.83e3 + 5.83e3i)T + 4.18e8iT^{2} \)
59 \( 1 + 3.17e4T + 7.14e8T^{2} \)
61 \( 1 - 2.52e4T + 8.44e8T^{2} \)
67 \( 1 + (-6.88e3 - 6.88e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 8.32e3iT - 1.80e9T^{2} \)
73 \( 1 + (-2.71e4 + 2.71e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.77e3iT - 3.07e9T^{2} \)
83 \( 1 + (-2.38e4 - 2.38e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.18e4T + 5.58e9T^{2} \)
97 \( 1 + (-6.95e4 - 6.95e4i)T + 8.58e9iT^{2} \)
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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

−9.537437585831380988965791906583, −8.961129241630922514745061271622, −8.255770813084390965676776067713, −7.25495282502065574298058401886, −6.37641853598521465679941120893, −5.30558486957665678669371020436, −4.73080853363414320171992059104, −3.39997730427159116691853720784, −2.37894912676795995824846460849, −1.35994649757109787376241208425, 0.04968316351244269336567164669, 1.26459951818602362948179044881, 2.29432972633481893231956273562, 3.52666648410790564570431529659, 4.72036541906003943084186093224, 5.07874308485817240315120530024, 6.56612923292885193959675822228, 7.28607399378005344786030109972, 7.996471275463885524983988945087, 8.868771948075253384254570304155

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