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Properties

Label 2-300-4.3-c4-0-71
Degree $2$
Conductor $300$
Sign $-0.993 - 0.117i$
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  + (2.99 − 2.65i)2-s + 5.19i·3-s + (1.88 − 15.8i)4-s + (13.8 + 15.5i)6-s − 74.3i·7-s + (−36.5 − 52.5i)8-s − 27·9-s + 107. i·11-s + (82.5 + 9.79i)12-s − 30.7·13-s + (−197. − 222. i)14-s + (−248. − 59.9i)16-s − 292.·17-s + (−80.7 + 71.7i)18-s − 357. i·19-s + ⋯
L(s)  = 1  + (0.747 − 0.664i)2-s + 0.577i·3-s + (0.117 − 0.993i)4-s + (0.383 + 0.431i)6-s − 1.51i·7-s + (−0.571 − 0.820i)8-s − 0.333·9-s + 0.887i·11-s + (0.573 + 0.0680i)12-s − 0.181·13-s + (−1.00 − 1.13i)14-s + (−0.972 − 0.234i)16-s − 1.01·17-s + (−0.249 + 0.221i)18-s − 0.991i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.993 - 0.117i$
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: \((2,\ 300,\ (\ :2),\ -0.993 - 0.117i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.286700462\)
\(L(\frac12)\) \(\approx\) \(1.286700462\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.99 + 2.65i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 + 74.3iT - 2.40e3T^{2} \)
11 \( 1 - 107. iT - 1.46e4T^{2} \)
13 \( 1 + 30.7T + 2.85e4T^{2} \)
17 \( 1 + 292.T + 8.35e4T^{2} \)
19 \( 1 + 357. iT - 1.30e5T^{2} \)
23 \( 1 - 488. iT - 2.79e5T^{2} \)
29 \( 1 + 1.03e3T + 7.07e5T^{2} \)
31 \( 1 - 411. iT - 9.23e5T^{2} \)
37 \( 1 - 1.50e3T + 1.87e6T^{2} \)
41 \( 1 + 2.00e3T + 2.82e6T^{2} \)
43 \( 1 + 3.27e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.60e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.38e3T + 7.89e6T^{2} \)
59 \( 1 - 851. iT - 1.21e7T^{2} \)
61 \( 1 + 5.53e3T + 1.38e7T^{2} \)
67 \( 1 + 4.18e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.30e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.03e3T + 2.83e7T^{2} \)
79 \( 1 - 7.01e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.12e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.18e3T + 6.27e7T^{2} \)
97 \( 1 - 1.22e4T + 8.85e7T^{2} \)
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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

−10.78389484182021870594780631669, −9.906805211603431794586266168574, −9.169605748369195586254289176283, −7.44298445477061905086567785369, −6.61836907895467324692769659898, −5.09465502970685831610425891633, −4.34184595849503647496598639413, −3.43899687741314432576409022869, −1.88812934339368012398108878825, −0.27988063976965514426961116440, 2.12813994024752465926210255015, 3.17555087020792339153567950306, 4.72000155663125814518173110181, 5.93709674805108108075656033939, 6.30001963189022984237791992183, 7.74579091992586455606787333949, 8.488768992008093764238406049060, 9.305565166473629838248837344390, 11.12995718354041516719705076464, 11.78560674999126560555651089043

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