Properties

Label 2-300-300.239-c1-0-37
Degree $2$
Conductor $300$
Sign $0.961 + 0.276i$
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  + (1.41 − 0.105i)2-s + (−1.72 − 0.0865i)3-s + (1.97 − 0.296i)4-s + (2.23 − 0.138i)5-s + (−2.44 + 0.0599i)6-s − 1.71·7-s + (2.75 − 0.627i)8-s + (2.98 + 0.299i)9-s + (3.13 − 0.429i)10-s + (3.14 − 2.28i)11-s + (−3.44 + 0.342i)12-s + (−2.67 + 3.68i)13-s + (−2.41 + 0.180i)14-s + (−3.87 + 0.0454i)15-s + (3.82 − 1.17i)16-s + (−1.44 − 4.44i)17-s + ⋯
L(s)  = 1  + (0.997 − 0.0744i)2-s + (−0.998 − 0.0499i)3-s + (0.988 − 0.148i)4-s + (0.998 − 0.0617i)5-s + (−0.999 + 0.0244i)6-s − 0.648·7-s + (0.975 − 0.221i)8-s + (0.995 + 0.0998i)9-s + (0.990 − 0.135i)10-s + (0.949 − 0.689i)11-s + (−0.995 + 0.0988i)12-s + (−0.741 + 1.02i)13-s + (−0.646 + 0.0482i)14-s + (−0.999 + 0.0117i)15-s + (0.955 − 0.293i)16-s + (−0.350 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.961 + 0.276i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.961 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98733 - 0.280180i\)
\(L(\frac12)\) \(\approx\) \(1.98733 - 0.280180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.105i)T \)
3 \( 1 + (1.72 + 0.0865i)T \)
5 \( 1 + (-2.23 + 0.138i)T \)
good7 \( 1 + 1.71T + 7T^{2} \)
11 \( 1 + (-3.14 + 2.28i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.67 - 3.68i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.44 + 4.44i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.75 + 0.895i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.94 - 5.43i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (7.46 + 2.42i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.34 - 2.06i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.44 - 4.74i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.75 - 5.16i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.64T + 43T^{2} \)
47 \( 1 + (5.75 + 1.86i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.134 + 0.415i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.61 + 1.90i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.20 + 3.78i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.75 + 5.41i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.57 - 10.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.32 - 5.95i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.64 - 1.18i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.21 - 1.36i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.46 - 3.38i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-8.74 - 2.84i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.52907015494989977701287798812, −11.32266779236897254336221477081, −9.834302875231139873883939704753, −9.355510313119154127390746056870, −7.09713345254413201331236015946, −6.65540816745262930810382788198, −5.60220218103222723096036953499, −4.85035605351830221651178146661, −3.37592763790948262233571594717, −1.65622392223979067766123258709, 1.85643348318077976476150478655, 3.57077769940985175516937924712, 4.92390575164288228945182996376, 5.74967977678704520078096838901, 6.58480396221537612135239518602, 7.34310565717193304107054787065, 9.262869138026554475132911691803, 10.25071813281444566855607308329, 10.84975415814976085385979970605, 12.07018508231573992889364472195

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