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Properties

Label 2-300-60.23-c2-0-61
Degree $2$
Conductor $300$
Sign $-0.315 + 0.948i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.961 + 1.75i)2-s + (0.903 − 2.86i)3-s + (−2.15 + 3.37i)4-s + (5.88 − 1.16i)6-s + (−7.30 − 7.30i)7-s + (−7.98 − 0.535i)8-s + (−7.36 − 5.17i)9-s − 4.41·11-s + (7.69 + 9.20i)12-s + (−7.53 − 7.53i)13-s + (5.78 − 19.8i)14-s + (−6.73 − 14.5i)16-s + (0.350 + 0.350i)17-s + (1.99 − 17.8i)18-s − 9.24·19-s + ⋯
L(s)  = 1  + (0.480 + 0.876i)2-s + (0.301 − 0.953i)3-s + (−0.538 + 0.842i)4-s + (0.981 − 0.193i)6-s + (−1.04 − 1.04i)7-s + (−0.997 − 0.0669i)8-s + (−0.818 − 0.574i)9-s − 0.401·11-s + (0.641 + 0.767i)12-s + (−0.579 − 0.579i)13-s + (0.413 − 1.41i)14-s + (−0.420 − 0.907i)16-s + (0.0206 + 0.0206i)17-s + (0.110 − 0.993i)18-s − 0.486·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.315 + 0.948i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ -0.315 + 0.948i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.525986 - 0.728998i\)
\(L(\frac12)\) \(\approx\) \(0.525986 - 0.728998i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.961 - 1.75i)T \)
3 \( 1 + (-0.903 + 2.86i)T \)
5 \( 1 \)
good7 \( 1 + (7.30 + 7.30i)T + 49iT^{2} \)
11 \( 1 + 4.41T + 121T^{2} \)
13 \( 1 + (7.53 + 7.53i)T + 169iT^{2} \)
17 \( 1 + (-0.350 - 0.350i)T + 289iT^{2} \)
19 \( 1 + 9.24T + 361T^{2} \)
23 \( 1 + (-17.9 - 17.9i)T + 529iT^{2} \)
29 \( 1 - 5.52T + 841T^{2} \)
31 \( 1 + 48.1iT - 961T^{2} \)
37 \( 1 + (3.39 - 3.39i)T - 1.36e3iT^{2} \)
41 \( 1 + 33.0iT - 1.68e3T^{2} \)
43 \( 1 + (-1.45 + 1.45i)T - 1.84e3iT^{2} \)
47 \( 1 + (27.8 - 27.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-52.6 + 52.6i)T - 2.80e3iT^{2} \)
59 \( 1 - 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 46.1T + 3.72e3T^{2} \)
67 \( 1 + (32.1 + 32.1i)T + 4.48e3iT^{2} \)
71 \( 1 + 116.T + 5.04e3T^{2} \)
73 \( 1 + (-72.2 - 72.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 55.9T + 6.24e3T^{2} \)
83 \( 1 + (46.5 + 46.5i)T + 6.88e3iT^{2} \)
89 \( 1 + 33.2T + 7.92e3T^{2} \)
97 \( 1 + (-24.6 + 24.6i)T - 9.40e3iT^{2} \)
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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

−11.54415268660763406190380067905, −10.13083506326286249331490985715, −9.114855803785565406393263698750, −7.935392333696195232031822288778, −7.27307979523518102307333378303, −6.51029931912016332819266049519, −5.45434833328027935118654714196, −3.89823793507940178444270542491, −2.81073709768130934507618217590, −0.33798447060367157276250503385, 2.43160014414355440909827325111, 3.23086251966830065802083558910, 4.55696151413889449859649712034, 5.45452011725756655613515473243, 6.59009455718044670846056311903, 8.575840310808209395173224709507, 9.192347728754831612999836451633, 10.02199469265154306498530557371, 10.76081940495054536567084472119, 11.85744986265124304037654686112

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