Properties

Label 2-300-12.11-c1-0-2
Degree $2$
Conductor $300$
Sign $0.273 - 0.961i$
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  + (−1.29 − 0.569i)2-s + (−0.908 − 1.47i)3-s + (1.35 + 1.47i)4-s + (0.335 + 2.42i)6-s + 2.50i·7-s + (−0.908 − 2.67i)8-s + (−1.35 + 2.67i)9-s − 3.36·11-s + (0.948 − 3.33i)12-s − 3.70·13-s + (1.42 − 3.24i)14-s + (−0.350 + 3.98i)16-s + 7.63i·17-s + (3.27 − 2.69i)18-s − 0.440i·19-s + ⋯
L(s)  = 1  + (−0.915 − 0.402i)2-s + (−0.524 − 0.851i)3-s + (0.675 + 0.737i)4-s + (0.136 + 0.990i)6-s + 0.948i·7-s + (−0.321 − 0.947i)8-s + (−0.450 + 0.892i)9-s − 1.01·11-s + (0.273 − 0.961i)12-s − 1.02·13-s + (0.382 − 0.868i)14-s + (−0.0876 + 0.996i)16-s + 1.85i·17-s + (0.771 − 0.635i)18-s − 0.100i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.273 - 0.961i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.273 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295555 + 0.223138i\)
\(L(\frac12)\) \(\approx\) \(0.295555 + 0.223138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.99310104648245425542062680758, −10.90035844669741893559451265413, −10.29458169522733644265770748146, −8.950659692556381359769357613694, −8.168916120863393892274265373458, −7.28481521658842475970704406953, −6.23290653187111138871373086109, −5.10488937975200745086128154332, −2.92526794406119207142279040446, −1.79824033868305525880523082077, 0.37040779619600545298043630020, 2.85796468558204773986180809540, 4.71501548691725087858585452637, 5.46504811515728168797790004881, 6.94094057728062594579758191906, 7.55054855015092814883072220977, 8.918744315837576878600919356855, 9.814974282656205383428285354516, 10.38174088947080060437316144887, 11.21967378310254057984271517228

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