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Properties

Label 2-30e2-5.2-c2-0-8
Degree $2$
Conductor $900$
Sign $0.850 + 0.525i$
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  + (1 + i)7-s − 15.8·11-s + (6 − 6i)13-s + (15.8 + 15.8i)17-s − 14i·19-s + (15.8 − 15.8i)23-s − 15.8i·29-s + 16·31-s + (30 + 30i)37-s + 31.6·41-s + (54 − 54i)43-s + (−47.4 − 47.4i)47-s − 47i·49-s + (−63.2 + 63.2i)53-s + 79.0i·59-s + ⋯
L(s)  = 1  + (0.142 + 0.142i)7-s − 1.43·11-s + (0.461 − 0.461i)13-s + (0.930 + 0.930i)17-s − 0.736i·19-s + (0.687 − 0.687i)23-s − 0.545i·29-s + 0.516·31-s + (0.810 + 0.810i)37-s + 0.771·41-s + (1.25 − 1.25i)43-s + (−1.00 − 1.00i)47-s − 0.959i·49-s + (−1.19 + 1.19i)53-s + 1.33i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.742045387\)
\(L(\frac12)\) \(\approx\) \(1.742045387\)
\(L(2)\) 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 \( 1 + (-1 - i)T + 49iT^{2} \)
11 \( 1 + 15.8T + 121T^{2} \)
13 \( 1 + (-6 + 6i)T - 169iT^{2} \)
17 \( 1 + (-15.8 - 15.8i)T + 289iT^{2} \)
19 \( 1 + 14iT - 361T^{2} \)
23 \( 1 + (-15.8 + 15.8i)T - 529iT^{2} \)
29 \( 1 + 15.8iT - 841T^{2} \)
31 \( 1 - 16T + 961T^{2} \)
37 \( 1 + (-30 - 30i)T + 1.36e3iT^{2} \)
41 \( 1 - 31.6T + 1.68e3T^{2} \)
43 \( 1 + (-54 + 54i)T - 1.84e3iT^{2} \)
47 \( 1 + (47.4 + 47.4i)T + 2.20e3iT^{2} \)
53 \( 1 + (63.2 - 63.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 79.0iT - 3.48e3T^{2} \)
61 \( 1 - 54T + 3.72e3T^{2} \)
67 \( 1 + (-34 - 34i)T + 4.48e3iT^{2} \)
71 \( 1 - 63.2T + 5.04e3T^{2} \)
73 \( 1 + (-65 + 65i)T - 5.32e3iT^{2} \)
79 \( 1 + 108iT - 6.24e3T^{2} \)
83 \( 1 + (-47.4 + 47.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 126. iT - 7.92e3T^{2} \)
97 \( 1 + (-69 - 69i)T + 9.40e3iT^{2} \)
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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

−10.05075941119847608551290988908, −8.874576478505670359884065958096, −8.129265342930364814969735320341, −7.47535872237438388877383291244, −6.27448797103342918570628434134, −5.46941069031287127105287705400, −4.58764731794192803271179355086, −3.29696460856281093164272371008, −2.33290479195413455745788471762, −0.71262810418280217787047583406, 1.02391828075681367834250441193, 2.51801254698134409589825625076, 3.52633155187517743274469051161, 4.79695281462318131155442300698, 5.52060556928902702599626362417, 6.54726669437684143149979597277, 7.73429414202443475624841178616, 7.993727905232078899373389133961, 9.355735817191130131539116871900, 9.848323793065984659111553125005

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