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Properties

Label 2-300-60.59-c1-0-23
Degree $2$
Conductor $300$
Sign $0.607 + 0.794i$
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  + (0.569 − 1.29i)2-s + (1.47 + 0.908i)3-s + (−1.35 − 1.47i)4-s + (2.01 − 1.39i)6-s + 2.50·7-s + (−2.67 + 0.908i)8-s + (1.35 + 2.67i)9-s + 3.36·11-s + (−0.652 − 3.40i)12-s − 3.70i·13-s + (1.42 − 3.24i)14-s + (−0.350 + 3.98i)16-s − 7.63·17-s + (4.23 − 0.222i)18-s + 0.440i·19-s + ⋯
L(s)  = 1  + (0.402 − 0.915i)2-s + (0.851 + 0.524i)3-s + (−0.675 − 0.737i)4-s + (0.822 − 0.568i)6-s + 0.948·7-s + (−0.947 + 0.321i)8-s + (0.450 + 0.892i)9-s + 1.01·11-s + (−0.188 − 0.982i)12-s − 1.02i·13-s + (0.382 − 0.868i)14-s + (−0.0876 + 0.996i)16-s − 1.85·17-s + (0.998 − 0.0523i)18-s + 0.100i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.607 + 0.794i$
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: \((2,\ 300,\ (\ :1/2),\ 0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82264 - 0.900218i\)
\(L(\frac12)\) \(\approx\) \(1.82264 - 0.900218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.569 + 1.29i)T \)
3 \( 1 + (-1.47 - 0.908i)T \)
5 \( 1 \)
good7 \( 1 - 2.50T + 7T^{2} \)
11 \( 1 - 3.36T + 11T^{2} \)
13 \( 1 + 3.70iT - 13T^{2} \)
17 \( 1 + 7.63T + 17T^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + 5.17iT - 23T^{2} \)
29 \( 1 - 2.27iT - 29T^{2} \)
31 \( 1 - 3.39iT - 31T^{2} \)
37 \( 1 - 7.40iT - 37T^{2} \)
41 \( 1 - 3.07iT - 41T^{2} \)
43 \( 1 + 8.40T + 43T^{2} \)
47 \( 1 + 3.63iT - 47T^{2} \)
53 \( 1 + 2.27T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 5.70T + 61T^{2} \)
67 \( 1 + 5.45T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 1.29iT - 73T^{2} \)
79 \( 1 - 5.01iT - 79T^{2} \)
83 \( 1 - 1.81iT - 83T^{2} \)
89 \( 1 - 5.35iT - 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−11.41999828628998102233879957323, −10.75976870926101744987840348135, −9.874637653476991316874216580593, −8.784964453305591091208800543809, −8.296951935880672673935739361222, −6.61591209045873692450644076789, −4.99921353127093429860591321037, −4.31126789135196370885223541914, −3.06218842420574496143559062842, −1.75914422939315494596899954358, 1.98774797936977134987421084730, 3.80185385809493529220651665430, 4.64133779493782875581534187107, 6.25672583487593629319581802784, 7.02382814018635405955561710775, 7.930559855986752643698091635589, 8.934981915733573855124986961638, 9.350896124197243605776520481692, 11.31159137161362425922714548210, 11.98478595820329485996961794504

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