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Properties

Label 2-300-4.3-c4-0-37
Degree $2$
Conductor $300$
Sign $0.417 + 0.908i$
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  + (−0.854 − 3.90i)2-s + 5.19i·3-s + (−14.5 + 6.67i)4-s + (20.3 − 4.44i)6-s − 10.5i·7-s + (38.5 + 51.1i)8-s − 27·9-s − 38.4i·11-s + (−34.7 − 75.5i)12-s − 273.·13-s + (−41.1 + 8.99i)14-s + (166. − 194. i)16-s + 256.·17-s + (23.0 + 105. i)18-s + 257. i·19-s + ⋯
L(s)  = 1  + (−0.213 − 0.976i)2-s + 0.577i·3-s + (−0.908 + 0.417i)4-s + (0.564 − 0.123i)6-s − 0.214i·7-s + (0.601 + 0.798i)8-s − 0.333·9-s − 0.317i·11-s + (−0.241 − 0.524i)12-s − 1.61·13-s + (−0.209 + 0.0458i)14-s + (0.651 − 0.758i)16-s + 0.887·17-s + (0.0712 + 0.325i)18-s + 0.712i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.417 + 0.908i$
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: \((2,\ 300,\ (\ :2),\ 0.417 + 0.908i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.293864941\)
\(L(\frac12)\) \(\approx\) \(1.293864941\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.854 + 3.90i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 + 10.5iT - 2.40e3T^{2} \)
11 \( 1 + 38.4iT - 1.46e4T^{2} \)
13 \( 1 + 273.T + 2.85e4T^{2} \)
17 \( 1 - 256.T + 8.35e4T^{2} \)
19 \( 1 - 257. iT - 1.30e5T^{2} \)
23 \( 1 + 168. iT - 2.79e5T^{2} \)
29 \( 1 - 684.T + 7.07e5T^{2} \)
31 \( 1 - 754. iT - 9.23e5T^{2} \)
37 \( 1 - 1.65e3T + 1.87e6T^{2} \)
41 \( 1 - 1.20e3T + 2.82e6T^{2} \)
43 \( 1 + 3.41e3iT - 3.41e6T^{2} \)
47 \( 1 + 3.24e3iT - 4.87e6T^{2} \)
53 \( 1 - 639.T + 7.89e6T^{2} \)
59 \( 1 - 6.77e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.88e3T + 1.38e7T^{2} \)
67 \( 1 + 6.75e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.21e3T + 2.83e7T^{2} \)
79 \( 1 + 7.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 5.47e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.60e3T + 6.27e7T^{2} \)
97 \( 1 + 758.T + 8.85e7T^{2} \)
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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

−10.66875014525600730899694167431, −10.16147999642685060976112461679, −9.337436396592985623036998474861, −8.309309076043488430334567872742, −7.31326085973671176555788896050, −5.55897027944886440570465501912, −4.58726921963777832496556477934, −3.47838374907805780711024314367, −2.33081110638012117355106491676, −0.61826838969108381564438733035, 0.854380733010977276700409014284, 2.60536928980615592976563735380, 4.46016837763835867575137506907, 5.44350617851782438855975745818, 6.51345383729322947517139986762, 7.47346861973126422366743110676, 8.051816057337038870039113479947, 9.384241166725972133340420587673, 9.905015673903158093848163290536, 11.33623182333834939238578206702

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