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Properties

Label 2-30e2-60.47-c0-0-0
Degree $2$
Conductor $900$
Sign $0.920 - 0.391i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (1 − i)13-s − 1.00·16-s + 1.41i·26-s + 1.41·29-s + (0.707 − 0.707i)32-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (−1.00 − 1.00i)52-s + (−1.00 + 1.00i)58-s + 1.00i·64-s + (−1 + i)73-s − 1.41·74-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (1 − i)13-s − 1.00·16-s + 1.41i·26-s + 1.41·29-s + (0.707 − 0.707i)32-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (−1.00 − 1.00i)52-s + (−1.00 + 1.00i)58-s + 1.00i·64-s + (−1 + i)73-s − 1.41·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.920 - 0.391i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.920 - 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7380269837\)
\(L(\frac12)\) \(\approx\) \(0.7380269837\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−10.31306200753245864371643572220, −9.438723652618452412335122686870, −8.503013309407695348192527299584, −8.023245722412576788004270925851, −7.00046957759112566052959720995, −6.14777995842846822443576016168, −5.40395388392271782895369443928, −4.27499228808693724421674533876, −2.81387087182950764640514526611, −1.17460419426996029119861987527, 1.35560445489355166254832841367, 2.62922627588755500792718818633, 3.78105983815928603619611147719, 4.64550398783395833253484122584, 6.15759558850442512729912174432, 6.96567193406411933816533276834, 8.008155191505087720301018702208, 8.703023568523237304613196399376, 9.445698443152276969134315994917, 10.24303806571410968256964277428

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