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Properties

Label 2-300-100.67-c1-0-15
Degree $2$
Conductor $300$
Sign $0.138 + 0.990i$
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  + (−1.36 − 0.352i)2-s + (−0.987 − 0.156i)3-s + (1.75 + 0.964i)4-s + (1.21 − 1.87i)5-s + (1.29 + 0.561i)6-s + (−0.466 + 0.466i)7-s + (−2.06 − 1.93i)8-s + (0.951 + 0.309i)9-s + (−2.32 + 2.14i)10-s + (1.78 − 0.578i)11-s + (−1.57 − 1.22i)12-s + (0.234 + 0.460i)13-s + (0.803 − 0.475i)14-s + (−1.49 + 1.66i)15-s + (2.14 + 3.37i)16-s + (−0.610 − 3.85i)17-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (−0.570 − 0.0903i)3-s + (0.876 + 0.482i)4-s + (0.543 − 0.839i)5-s + (0.529 + 0.229i)6-s + (−0.176 + 0.176i)7-s + (−0.728 − 0.685i)8-s + (0.317 + 0.103i)9-s + (−0.735 + 0.677i)10-s + (0.537 − 0.174i)11-s + (−0.456 − 0.354i)12-s + (0.0650 + 0.127i)13-s + (0.214 − 0.126i)14-s + (−0.385 + 0.429i)15-s + (0.535 + 0.844i)16-s + (−0.147 − 0.934i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.138 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554507 - 0.482317i\)
\(L(\frac12)\) \(\approx\) \(0.554507 - 0.482317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.352i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (-1.21 + 1.87i)T \)
good7 \( 1 + (0.466 - 0.466i)T - 7iT^{2} \)
11 \( 1 + (-1.78 + 0.578i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.234 - 0.460i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.610 + 3.85i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.124 + 0.0902i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.60 + 5.12i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (2.19 + 3.02i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.91 + 6.77i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.232 + 0.118i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.02 + 9.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + 43iT^{2} \)
47 \( 1 + (1.19 - 7.53i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.540 - 3.41i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (4.23 - 13.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.806 + 2.48i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.495 + 0.0784i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-3.34 - 4.60i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-14.3 - 7.30i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-4.18 + 3.03i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.77 + 11.2i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (17.1 - 5.56i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (-13.4 - 2.12i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.49966241214826538370819780034, −10.53200561514186995498769665443, −9.491868937821966068224272471942, −8.978614862191804417961061584067, −7.82817989144720473698425149646, −6.64085475487371126511570134041, −5.78113775450789858186329394045, −4.34558967437631045904158493636, −2.44543447410694564407276695040, −0.843856473286204930734298632330, 1.63226931593824835597832780676, 3.34626940733509012793705342270, 5.28383407152246295273214380644, 6.40580532272985885419063098434, 6.90865245764815487299529451442, 8.122288563394042076715243640191, 9.361217749699988153121529311620, 10.05976318729111121177075132669, 10.87019240972684632046026103831, 11.52938704747279820873789596807

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