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Properties

Label 2-300-15.2-c1-0-5
Degree $2$
Conductor $300$
Sign $0.229 + 0.973i$
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 − 1.22i)3-s + (−2.44 − 2.44i)7-s − 2.99i·9-s + (4.89 − 4.89i)13-s + 8i·19-s − 5.99·21-s + (−3.67 − 3.67i)27-s + 4·31-s + (4.89 + 4.89i)37-s − 11.9i·39-s + (−7.34 + 7.34i)43-s + 4.99i·49-s + (9.79 + 9.79i)57-s + 14·61-s + (−7.34 + 7.34i)63-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.925 − 0.925i)7-s − 0.999i·9-s + (1.35 − 1.35i)13-s + 1.83i·19-s − 1.30·21-s + (−0.707 − 0.707i)27-s + 0.718·31-s + (0.805 + 0.805i)37-s − 1.92i·39-s + (−1.12 + 1.12i)43-s + 0.714i·49-s + (1.29 + 1.29i)57-s + 1.79·61-s + (−0.925 + 0.925i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17066 - 0.926483i\)
\(L(\frac12)\) \(\approx\) \(1.17066 - 0.926483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.89 + 4.89i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (7.34 - 7.34i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + (2.44 + 2.44i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (9.79 - 9.79i)T - 73iT^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-9.79 - 9.79i)T + 97iT^{2} \)
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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

−11.66766425637560411303034890671, −10.37110439937222862078897240559, −9.794838228001785096407611600209, −8.395828040338956258713311183158, −7.86339715024447441911602339562, −6.65378478733536299864913578006, −5.89180492702692051383416284098, −3.87033357540568871028019178587, −3.10809987766291724972668468928, −1.14306251508691973050411521430, 2.34826722357148605880232686547, 3.51011191800166728322480057009, 4.67163413279996298194903414777, 6.02158778135602199514818276050, 7.03338182723814703896016506530, 8.628485723260140079931619149430, 8.998034393088949535561727581637, 9.834070822083326400729409964486, 11.01320479877194739921987132597, 11.77429025792328301980138238785

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