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Properties

Label 2-300-60.59-c1-0-9
Degree $2$
Conductor $300$
Sign $0.633 - 0.773i$
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  + (1.28 + 0.599i)2-s + (−1.66 − 0.468i)3-s + (1.28 + 1.53i)4-s + (−1.85 − 1.59i)6-s + 0.936·7-s + (0.719 + 2.73i)8-s + (2.56 + 1.56i)9-s + 4.27·11-s + (−1.41 − 3.16i)12-s + 3.12i·13-s + (1.19 + 0.561i)14-s + (−0.719 + 3.93i)16-s − 2·17-s + (2.34 + 3.53i)18-s + 4.27i·19-s + ⋯
L(s)  = 1  + (0.905 + 0.424i)2-s + (−0.962 − 0.270i)3-s + (0.640 + 0.768i)4-s + (−0.757 − 0.653i)6-s + 0.353·7-s + (0.254 + 0.967i)8-s + (0.853 + 0.520i)9-s + 1.28·11-s + (−0.408 − 0.912i)12-s + 0.866i·13-s + (0.320 + 0.150i)14-s + (−0.179 + 0.983i)16-s − 0.485·17-s + (0.552 + 0.833i)18-s + 0.979i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.633 - 0.773i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.633 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59284 + 0.754673i\)
\(L(\frac12)\) \(\approx\) \(1.59284 + 0.754673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.599i)T \)
3 \( 1 + (1.66 + 0.468i)T \)
5 \( 1 \)
good7 \( 1 - 0.936T + 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 3.12iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4.27iT - 19T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 + 2.39iT - 31T^{2} \)
37 \( 1 - 3.12iT - 37T^{2} \)
41 \( 1 + 7.12iT - 41T^{2} \)
43 \( 1 + 1.46T + 43T^{2} \)
47 \( 1 + 0.936iT - 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 + 5.12T + 61T^{2} \)
67 \( 1 - 5.20T + 67T^{2} \)
71 \( 1 + 6.67T + 71T^{2} \)
73 \( 1 + 8.24iT - 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 - 4.68iT - 83T^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−11.90742867975298529911918878749, −11.38957309182059550509579831156, −10.32341095358640392658907795220, −8.865331867399318177560110091381, −7.68175144725541460856449772098, −6.58749410091420625823324884529, −6.12156329310307562184429875377, −4.74406889480368944187998791227, −4.00486847023652414018317593202, −1.91135949239542909448822821589, 1.35654984211100453819360860612, 3.38033596623352599878866711225, 4.53197446742712637270205907944, 5.41631948138530771150600300652, 6.41940448967905031473483402226, 7.30805317373931707464100729373, 9.114979359764907926981935509804, 10.03216005545055388847836845015, 11.13602372963175475247766043735, 11.45692986944874225285668794800

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