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Properties

Label 2-300-4.3-c4-0-0
Degree $2$
Conductor $300$
Sign $-0.606 + 0.795i$
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  + (−1.28 + 3.78i)2-s − 5.19i·3-s + (−12.7 − 9.70i)4-s + (19.6 + 6.65i)6-s + 89.0i·7-s + (53.0 − 35.7i)8-s − 27·9-s − 174. i·11-s + (−50.4 + 66.1i)12-s + 22.9·13-s + (−337. − 114. i)14-s + (67.7 + 246. i)16-s − 69.2·17-s + (34.5 − 102. i)18-s + 341. i·19-s + ⋯
L(s)  = 1  + (−0.320 + 0.947i)2-s − 0.577i·3-s + (−0.795 − 0.606i)4-s + (0.546 + 0.184i)6-s + 1.81i·7-s + (0.828 − 0.559i)8-s − 0.333·9-s − 1.44i·11-s + (−0.350 + 0.459i)12-s + 0.136·13-s + (−1.72 − 0.581i)14-s + (0.264 + 0.964i)16-s − 0.239·17-s + (0.106 − 0.315i)18-s + 0.944i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.606 + 0.795i$
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: \((2,\ 300,\ (\ :2),\ -0.606 + 0.795i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.02276098926\)
\(L(\frac12)\) \(\approx\) \(0.02276098926\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 3.78i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 89.0iT - 2.40e3T^{2} \)
11 \( 1 + 174. iT - 1.46e4T^{2} \)
13 \( 1 - 22.9T + 2.85e4T^{2} \)
17 \( 1 + 69.2T + 8.35e4T^{2} \)
19 \( 1 - 341. iT - 1.30e5T^{2} \)
23 \( 1 - 319. iT - 2.79e5T^{2} \)
29 \( 1 - 679.T + 7.07e5T^{2} \)
31 \( 1 - 72.5iT - 9.23e5T^{2} \)
37 \( 1 + 2.37e3T + 1.87e6T^{2} \)
41 \( 1 + 762.T + 2.82e6T^{2} \)
43 \( 1 + 3.11e3iT - 3.41e6T^{2} \)
47 \( 1 + 315. iT - 4.87e6T^{2} \)
53 \( 1 + 3.38e3T + 7.89e6T^{2} \)
59 \( 1 + 6.68e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.31e3T + 1.38e7T^{2} \)
67 \( 1 + 4.01e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.95e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.74e3T + 2.83e7T^{2} \)
79 \( 1 - 414. iT - 3.89e7T^{2} \)
83 \( 1 - 9.73e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.19e3T + 6.27e7T^{2} \)
97 \( 1 + 9.56e3T + 8.85e7T^{2} \)
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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

−11.84838161466726236441338310056, −10.72615511587502281537102486714, −9.400827938791294510811960141473, −8.564099995862248642075635553031, −8.149098344286388231474535165524, −6.69964662612869379051768490993, −5.86201201785906956858390953375, −5.26326821367428326124826658151, −3.31601863510422575802153632640, −1.69814727461648609327160061324, 0.008096590414680710627243170113, 1.41860028081020085546024184171, 3.02150057974681659085221160655, 4.32619074219774044066545618099, 4.68430914913902880030871992804, 6.84367830268200173486926615611, 7.67560243739072656047020994823, 8.887588049488343043382782173793, 9.915589647604237132233827221515, 10.40631415927259081853367264510

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