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Properties

Label 2-300-100.23-c1-0-24
Degree $2$
Conductor $300$
Sign $-0.997 + 0.0724i$
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  + (−1.38 − 0.308i)2-s + (−0.453 − 0.891i)3-s + (1.80 + 0.851i)4-s + (−0.600 − 2.15i)5-s + (0.351 + 1.36i)6-s + (−0.972 − 0.972i)7-s + (−2.23 − 1.73i)8-s + (−0.587 + 0.809i)9-s + (0.164 + 3.15i)10-s + (−0.162 − 0.223i)11-s + (−0.0623 − 1.99i)12-s + (0.0801 + 0.505i)13-s + (1.04 + 1.64i)14-s + (−1.64 + 1.51i)15-s + (2.54 + 3.08i)16-s + (−4.63 − 2.36i)17-s + ⋯
L(s)  = 1  + (−0.975 − 0.218i)2-s + (−0.262 − 0.514i)3-s + (0.904 + 0.425i)4-s + (−0.268 − 0.963i)5-s + (0.143 + 0.559i)6-s + (−0.367 − 0.367i)7-s + (−0.789 − 0.613i)8-s + (−0.195 + 0.269i)9-s + (0.0519 + 0.998i)10-s + (−0.0488 − 0.0672i)11-s + (−0.0179 − 0.577i)12-s + (0.0222 + 0.140i)13-s + (0.278 + 0.438i)14-s + (−0.425 + 0.390i)15-s + (0.637 + 0.770i)16-s + (−1.12 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.997 + 0.0724i$
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: \((2,\ 300,\ (\ :1/2),\ -0.997 + 0.0724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0141139 - 0.388918i\)
\(L(\frac12)\) \(\approx\) \(0.0141139 - 0.388918i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.308i)T \)
3 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (0.600 + 2.15i)T \)
good7 \( 1 + (0.972 + 0.972i)T + 7iT^{2} \)
11 \( 1 + (0.162 + 0.223i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.0801 - 0.505i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (4.63 + 2.36i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.118 + 0.363i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.0164 + 0.104i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (5.13 + 1.66i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.03 - 1.31i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.32 - 1.47i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (0.259 + 0.188i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.23 + 6.23i)T - 43iT^{2} \)
47 \( 1 + (2.85 - 1.45i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-8.32 + 4.24i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-11.0 - 8.03i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.38 + 6.81i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.33 + 2.61i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (11.5 + 3.74i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.09 - 0.490i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-4.72 + 14.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-12.6 - 6.42i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (2.29 + 3.16i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.80 - 11.3i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.33188107737573541502382193132, −10.33950482274037330600346014103, −9.205749949899832557155787927409, −8.589841815656104982962751632659, −7.46667592131154205985144695451, −6.72303603717028832362063469516, −5.36900102682846244748515975972, −3.79544620392144295301018374701, −2.01142988859785701048924117101, −0.37935498504401293722485363692, 2.35372032613910643786706186136, 3.73816495863657343192581239895, 5.54515265690696341112036652948, 6.50190535245905326645920542422, 7.35111592594881175020523773824, 8.546174490129596363355609093060, 9.415396553507210440766087655800, 10.37564059875295790657739839298, 10.98867931843044135609831239753, 11.75322018400945105791559315279

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