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Properties

Label 2-300-4.3-c4-0-72
Degree $2$
Conductor $300$
Sign $-0.892 - 0.452i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.40 − 2.09i)2-s − 5.19i·3-s + (7.23 − 14.2i)4-s + (−10.8 − 17.7i)6-s + 61.3i·7-s + (−5.23 − 63.7i)8-s − 27·9-s − 74.1i·11-s + (−74.1 − 37.5i)12-s − 181.·13-s + (128. + 209. i)14-s + (−151. − 206. i)16-s − 516.·17-s + (−92.0 + 56.5i)18-s − 407. i·19-s + ⋯
L(s)  = 1  + (0.852 − 0.523i)2-s − 0.577i·3-s + (0.452 − 0.892i)4-s + (−0.302 − 0.491i)6-s + 1.25i·7-s + (−0.0817 − 0.996i)8-s − 0.333·9-s − 0.612i·11-s + (−0.515 − 0.260i)12-s − 1.07·13-s + (0.655 + 1.06i)14-s + (−0.591 − 0.806i)16-s − 1.78·17-s + (−0.284 + 0.174i)18-s − 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.452i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.892 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.892 - 0.452i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ -0.892 - 0.452i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.202294254\)
\(L(\frac12)\) \(\approx\) \(1.202294254\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.40 + 2.09i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 61.3iT - 2.40e3T^{2} \)
11 \( 1 + 74.1iT - 1.46e4T^{2} \)
13 \( 1 + 181.T + 2.85e4T^{2} \)
17 \( 1 + 516.T + 8.35e4T^{2} \)
19 \( 1 + 407. iT - 1.30e5T^{2} \)
23 \( 1 + 7.48iT - 2.79e5T^{2} \)
29 \( 1 + 1.47e3T + 7.07e5T^{2} \)
31 \( 1 + 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 - 667.T + 1.87e6T^{2} \)
41 \( 1 - 1.21e3T + 2.82e6T^{2} \)
43 \( 1 - 987. iT - 3.41e6T^{2} \)
47 \( 1 - 2.94e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.28e3T + 7.89e6T^{2} \)
59 \( 1 - 390. iT - 1.21e7T^{2} \)
61 \( 1 - 4.10e3T + 1.38e7T^{2} \)
67 \( 1 + 6.16e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.46e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.36e3T + 2.83e7T^{2} \)
79 \( 1 - 7.06e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.97e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.23e4T + 6.27e7T^{2} \)
97 \( 1 + 2.51e3T + 8.85e7T^{2} \)
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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

−11.22617329729790065182213075964, −9.549218752856066203148148718304, −8.882704469392593208540305619887, −7.41446979083306866242309943813, −6.34224140094378997195968993571, −5.51197649125103338573957121406, −4.41309893061003184065695021821, −2.74522486410990168069541734758, −2.11251130524515090331436818432, −0.24373607291584639192899211900, 2.20657685003600808325649628184, 3.79604742012603467127829849909, 4.42645526948045303284525948271, 5.48332968281094518507040459456, 6.87845208337097482011571349945, 7.43050565561892055235495027322, 8.673089068024885837696318505157, 9.897689970985632994405325305992, 10.77313856468064678138985750566, 11.68188985278256608961611625540

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