Properties

Label 2-300-20.7-c1-0-5
Degree $2$
Conductor $300$
Sign $0.755 - 0.655i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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 + (−0.707 − 0.707i)3-s + (0.999 − 1.73i)4-s + (1.36 + 0.366i)6-s + (−1.74 + 1.74i)7-s + 2.82i·8-s + 1.00i·9-s + 2i·11-s + (−1.93 + 0.517i)12-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 + (−0.707 − 1.22i)18-s + 7.19·19-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)2-s + (−0.408 − 0.408i)3-s + (0.499 − 0.866i)4-s + (0.557 + 0.149i)6-s + (−0.658 + 0.658i)7-s + 0.999i·8-s + 0.333i·9-s + 0.603i·11-s + (−0.557 + 0.149i)12-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 + (−0.166 − 0.288i)18-s + 1.65·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.755 - 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704284 + 0.262963i\)
\(L(\frac12)\) \(\approx\) \(0.704284 + 0.262963i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
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

−11.82925061662946232355446989120, −10.65104952117280602504545143032, −9.995744338858016055722009257712, −8.917929458032088371801583673515, −7.993691309437563466282345128106, −7.08075961985905466094740085119, −5.94631669678817738058631871140, −5.39966334786721538441851554985, −3.15591942293845268274111660768, −1.30192589175606002282227612411, 0.953076200547883476621943553750, 3.12776389750713360144645410640, 4.06911851531989675583786129916, 5.82108354802297532821676286716, 6.91636847323786099669323612358, 7.86613859997766633086490789798, 9.203690835129743405870333414419, 9.685764237209467808405558334846, 10.66915712388318635495114202680, 11.53134728088508847738641066820

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