Properties

Label 2-300-75.38-c1-0-2
Degree $2$
Conductor $300$
Sign $0.996 + 0.0860i$
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  + (−1.73 + 0.0417i)3-s + (2.12 − 0.707i)5-s + (−1.40 + 1.40i)7-s + (2.99 − 0.144i)9-s + (1.16 − 1.60i)11-s + (1.08 + 0.172i)13-s + (−3.64 + 1.31i)15-s + (3.88 − 1.98i)17-s + (7.34 + 2.38i)19-s + (2.38 − 2.49i)21-s + (−0.596 + 0.0944i)23-s + (3.99 − 3.00i)25-s + (−5.18 + 0.375i)27-s + (1.20 + 3.70i)29-s + (2.08 − 6.41i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0241i)3-s + (0.948 − 0.316i)5-s + (−0.532 + 0.532i)7-s + (0.998 − 0.0482i)9-s + (0.352 − 0.484i)11-s + (0.301 + 0.0477i)13-s + (−0.940 + 0.339i)15-s + (0.942 − 0.480i)17-s + (1.68 + 0.547i)19-s + (0.519 − 0.545i)21-s + (−0.124 + 0.0196i)23-s + (0.799 − 0.600i)25-s + (−0.997 + 0.0723i)27-s + (0.223 + 0.688i)29-s + (0.374 − 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.996 + 0.0860i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.996 + 0.0860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13826 - 0.0490514i\)
\(L(\frac12)\) \(\approx\) \(1.13826 - 0.0490514i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.73 - 0.0417i)T \)
5 \( 1 + (-2.12 + 0.707i)T \)
good7 \( 1 + (1.40 - 1.40i)T - 7iT^{2} \)
11 \( 1 + (-1.16 + 1.60i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.08 - 0.172i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.88 + 1.98i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-7.34 - 2.38i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.596 - 0.0944i)T + (21.8 - 7.10i)T^{2} \)
29 \( 1 + (-1.20 - 3.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.08 + 6.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.00 - 6.36i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (4.90 + 6.75i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.08 - 2.08i)T + 43iT^{2} \)
47 \( 1 + (1.27 - 2.50i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (9.48 + 4.83i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (11.0 - 8.00i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.73 + 5.62i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (5.76 + 11.3i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (-2.90 + 0.942i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.52 - 9.64i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-8.77 + 2.85i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.61 - 3.16i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + (-2.65 - 1.93i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.19 - 0.611i)T + (57.0 + 78.4i)T^{2} \)
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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

−11.91582816221788749074114415912, −10.79012160468056163308923672093, −9.756967870902527559696798380928, −9.298600682766895261368045546591, −7.80039792093437775242327306562, −6.46514923030704055549276345615, −5.79099223858607943313499580659, −4.97276367247928942284916654502, −3.24585274161497203935392865132, −1.28809682736206424544958501300, 1.35015309821522047566022482130, 3.33510048574463091889452027592, 4.85879824195835460741725015641, 5.87956672738476795483691190637, 6.70259334069062851625306876488, 7.58189278054551631380933672345, 9.332347629261535404838983168446, 10.01100101871147625781790114285, 10.68260577258784944254772734804, 11.76883792869686536682644529345

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