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Properties

Label 2-30e2-20.7-c1-0-18
Degree $2$
Conductor $900$
Sign $0.0706 - 0.997i$
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  + (0.707 + 1.22i)2-s + (−0.999 + 1.73i)4-s + (3.15 − 3.15i)7-s − 2.82·8-s + 2i·11-s + (−1.60 + 1.60i)13-s + (6.09 + 1.63i)14-s + (−2.00 − 3.46i)16-s + (4.24 + 4.24i)17-s + 3.19·19-s + (−2.44 + 1.41i)22-s + (5.27 + 5.27i)23-s + (−3.09 − 0.830i)26-s + (2.31 + 8.62i)28-s − 0.535i·29-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.19 − 1.19i)7-s − 0.999·8-s + 0.603i·11-s + (−0.444 + 0.444i)13-s + (1.62 + 0.436i)14-s + (−0.500 − 0.866i)16-s + (1.02 + 1.02i)17-s + 0.733·19-s + (−0.522 + 0.301i)22-s + (1.10 + 1.10i)23-s + (−0.607 − 0.162i)26-s + (0.436 + 1.62i)28-s − 0.0995i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0706 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0706 - 0.997i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.0706 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58507 + 1.47678i\)
\(L(\frac12)\) \(\approx\) \(1.58507 + 1.47678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-3.15 + 3.15i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (1.60 - 1.60i)T - 13iT^{2} \)
17 \( 1 + (-4.24 - 4.24i)T + 17iT^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (-5.27 - 5.27i)T + 23iT^{2} \)
29 \( 1 + 0.535iT - 29T^{2} \)
31 \( 1 + 3.73iT - 31T^{2} \)
37 \( 1 + (-7.72 - 7.72i)T + 37iT^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + (3.15 + 3.15i)T + 43iT^{2} \)
47 \( 1 + (-0.656 + 0.656i)T - 47iT^{2} \)
53 \( 1 + (3.86 - 3.86i)T - 53iT^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (3.91 - 3.91i)T - 67iT^{2} \)
71 \( 1 + 2.53iT - 71T^{2} \)
73 \( 1 + (-2.82 + 2.82i)T - 73iT^{2} \)
79 \( 1 + 7.46T + 79T^{2} \)
83 \( 1 + (9.14 + 9.14i)T + 83iT^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-6.88 - 6.88i)T + 97iT^{2} \)
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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

−10.19861747219102631460624516807, −9.422122458023875934456067294072, −8.255266703641929528006023432002, −7.57366521026455089264050515360, −7.14787700284042573399251178518, −5.93265390523191894724758117461, −4.92491319664439879674087075029, −4.32494862524979995805634313730, −3.27001426826099754077322021629, −1.43339545034444311987284133481, 1.06485092757788543471753680275, 2.47267256130478886305707750355, 3.19989935014388038122062766627, 4.77074473155397209506932692463, 5.22644472518426827785907919545, 6.05066686236567187931975031071, 7.48468926065016696249922667005, 8.465638638488833080754807860690, 9.160464524327743639672941257707, 9.990413765652615110876716526611

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