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Properties

Label 2-30e2-20.7-c1-0-38
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  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + (−1.74 + 1.74i)7-s − 2.82i·8-s − 2i·11-s + (4.05 − 4.05i)13-s + (−0.901 + 3.36i)14-s + (−2.00 − 3.46i)16-s + (−4.24 − 4.24i)17-s + 7.19·19-s + (−1.41 − 2.44i)22-s + (0.378 + 0.378i)23-s + (2.09 − 7.83i)26-s + (1.27 + 4.76i)28-s − 7.46i·29-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + (−0.658 + 0.658i)7-s − 0.999i·8-s − 0.603i·11-s + (1.12 − 1.12i)13-s + (−0.241 + 0.899i)14-s + (−0.500 − 0.866i)16-s + (−1.02 − 1.02i)17-s + 1.65·19-s + (−0.301 − 0.522i)22-s + (0.0790 + 0.0790i)23-s + (0.411 − 1.53i)26-s + (0.241 + 0.899i)28-s − 1.38i·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

\(L(1)\) \(\approx\) \(1.65724 - 1.77876i\)
\(L(\frac12)\) \(\approx\) \(1.65724 - 1.77876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (1.74 - 1.74i)T - 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + (-4.05 + 4.05i)T - 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 - 7.19T + 19T^{2} \)
23 \( 1 + (-0.378 - 0.378i)T + 23iT^{2} \)
29 \( 1 + 7.46iT - 29T^{2} \)
31 \( 1 - 0.267iT - 31T^{2} \)
37 \( 1 + (-2.07 - 2.07i)T + 37iT^{2} \)
41 \( 1 - 5.46T + 41T^{2} \)
43 \( 1 + (-1.74 - 1.74i)T + 43iT^{2} \)
47 \( 1 + (9.14 - 9.14i)T - 47iT^{2} \)
53 \( 1 + (1.03 - 1.03i)T - 53iT^{2} \)
59 \( 1 - 4.53T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + (8.81 - 8.81i)T - 67iT^{2} \)
71 \( 1 - 9.46iT - 71T^{2} \)
73 \( 1 + (2.82 - 2.82i)T - 73iT^{2} \)
79 \( 1 - 0.535T + 79T^{2} \)
83 \( 1 + (-0.656 - 0.656i)T + 83iT^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + (4.43 + 4.43i)T + 97iT^{2} \)
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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

−9.869304719434102774612699461930, −9.360205832189505050486444062709, −8.249532029049874699482531388661, −7.12435949603902789236582942069, −6.01905727320318914824448939688, −5.65477649977064011410741720199, −4.46973997245023848891776974284, −3.24369596551973188111339111936, −2.70462178886949743011580928214, −0.921786960479907610881697494765, 1.78890961001234894095448994371, 3.34573903607270297299181285717, 4.01854176721297901638358410990, 4.98549972481829787209503030427, 6.15192494177841170433652269748, 6.78801868287903410171563732091, 7.46388858017251548632090985196, 8.590260147129270991870453819552, 9.350049299577227273216653607265, 10.50192813460811353988754869507

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