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Properties

Label 2-300-300.239-c1-0-36
Degree $2$
Conductor $300$
Sign $0.737 + 0.674i$
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  + (−1.30 + 0.545i)2-s + (0.0653 − 1.73i)3-s + (1.40 − 1.42i)4-s + (2.14 − 0.638i)5-s + (0.858 + 2.29i)6-s + 4.06·7-s + (−1.05 + 2.62i)8-s + (−2.99 − 0.226i)9-s + (−2.44 + 2.00i)10-s + (2.61 − 1.89i)11-s + (−2.37 − 2.52i)12-s + (−3.55 + 4.88i)13-s + (−5.30 + 2.21i)14-s + (−0.964 − 3.75i)15-s + (−0.0500 − 3.99i)16-s + (0.528 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.922 + 0.385i)2-s + (0.0377 − 0.999i)3-s + (0.702 − 0.711i)4-s + (0.958 − 0.285i)5-s + (0.350 + 0.936i)6-s + 1.53·7-s + (−0.373 + 0.927i)8-s + (−0.997 − 0.0754i)9-s + (−0.774 + 0.632i)10-s + (0.787 − 0.571i)11-s + (−0.684 − 0.729i)12-s + (−0.985 + 1.35i)13-s + (−1.41 + 0.592i)14-s + (−0.249 − 0.968i)15-s + (−0.0125 − 0.999i)16-s + (0.128 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.737 + 0.674i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.737 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06536 - 0.413653i\)
\(L(\frac12)\) \(\approx\) \(1.06536 - 0.413653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.545i)T \)
3 \( 1 + (-0.0653 + 1.73i)T \)
5 \( 1 + (-2.14 + 0.638i)T \)
good7 \( 1 - 4.06T + 7T^{2} \)
11 \( 1 + (-2.61 + 1.89i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.55 - 4.88i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.528 - 1.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.49 + 0.809i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.71 + 3.74i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.01 + 0.655i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.77 - 0.576i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.36 - 1.88i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.31 + 5.94i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.20T + 43T^{2} \)
47 \( 1 + (3.27 + 1.06i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.20 - 6.78i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.81 + 7.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.99 - 3.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.983 - 3.02i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.67 - 5.15i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.58 + 2.17i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.42 - 0.787i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.77 + 1.87i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-4.38 - 6.04i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (14.9 + 4.85i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.61866347230265368028017627712, −10.73052147474612527655158903393, −9.397841809515323803439984903237, −8.744092482295189469038866957511, −7.85280243717266219348098893613, −6.87536248038973012899697551096, −5.97816361210132977336504096199, −4.89991796265983312486413215353, −2.19049558636361178649020233491, −1.39780777107537827318485333063, 1.78650597534645790430676585432, 3.14518865882686684255578915272, 4.74893234068154781849177119205, 5.76089092294585836855506713043, 7.37664974419485126499475025666, 8.195428434117604814578282370728, 9.406079359310680438343341371148, 9.850327904227237148921895855110, 10.75745716459411291872097050894, 11.47771587537596281258601385172

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