Properties

Label 2-300-100.23-c1-0-28
Degree $2$
Conductor $300$
Sign $-0.0310 + 0.999i$
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  + (1.40 − 0.155i)2-s + (−0.453 − 0.891i)3-s + (1.95 − 0.438i)4-s + (−1.39 − 1.74i)5-s + (−0.777 − 1.18i)6-s + (−2.75 − 2.75i)7-s + (2.67 − 0.920i)8-s + (−0.587 + 0.809i)9-s + (−2.23 − 2.23i)10-s + (−0.189 − 0.260i)11-s + (−1.27 − 1.53i)12-s + (0.0568 + 0.359i)13-s + (−4.30 − 3.44i)14-s + (−0.920 + 2.03i)15-s + (3.61 − 1.71i)16-s + (6.81 + 3.47i)17-s + ⋯
L(s)  = 1  + (0.993 − 0.110i)2-s + (−0.262 − 0.514i)3-s + (0.975 − 0.219i)4-s + (−0.624 − 0.780i)5-s + (−0.317 − 0.482i)6-s + (−1.04 − 1.04i)7-s + (0.945 − 0.325i)8-s + (−0.195 + 0.269i)9-s + (−0.707 − 0.706i)10-s + (−0.0571 − 0.0786i)11-s + (−0.368 − 0.444i)12-s + (0.0157 + 0.0996i)13-s + (−1.15 − 0.921i)14-s + (−0.237 + 0.526i)15-s + (0.903 − 0.427i)16-s + (1.65 + 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0310 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0310 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.0310 + 0.999i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.0310 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26886 - 1.30883i\)
\(L(\frac12)\) \(\approx\) \(1.26886 - 1.30883i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.155i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (1.39 + 1.74i)T \)
good7 \( 1 + (2.75 + 2.75i)T + 7iT^{2} \)
11 \( 1 + (0.189 + 0.260i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.0568 - 0.359i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-6.81 - 3.47i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.52 - 4.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.39 + 8.81i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-6.53 - 2.12i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.16 - 1.67i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.02 + 0.795i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (7.28 + 5.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.45 - 1.45i)T - 43iT^{2} \)
47 \( 1 + (0.891 - 0.454i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (2.04 - 1.04i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.88 - 4.27i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.66 - 2.66i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.984 - 1.93i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-8.42 - 2.73i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (12.8 + 2.03i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (3.45 - 10.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.05 - 2.06i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (2.42 + 3.34i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.13 - 2.22i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.96358111610879142322123818784, −10.64926846600765412295542331163, −10.03896274397952712032865031211, −8.334424070876280607177400400639, −7.40735041395362024522487747050, −6.49225339405775296078532636520, −5.45212617526080450049582903117, −4.17854849807319385822279371700, −3.27155095951207691237283644863, −1.12419459183531281518870214919, 2.90682401022535447263436453061, 3.43125079649912695344076199040, 5.00383878890185290839703841796, 5.88617855642752564145424247161, 6.88786795758163298954061565734, 7.83936325119684890786432399243, 9.412763722820051453597710815206, 10.20705927419481616499154645251, 11.56422601176542091629686705043, 11.73457333833730437875447783645

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