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Properties

Label 2-300-100.67-c1-0-19
Degree $2$
Conductor $300$
Sign $-0.950 + 0.311i$
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  + (−1.12 − 0.863i)2-s + (−0.987 − 0.156i)3-s + (0.508 + 1.93i)4-s + (−1.70 + 1.44i)5-s + (0.971 + 1.02i)6-s + (1.87 − 1.87i)7-s + (1.10 − 2.60i)8-s + (0.951 + 0.309i)9-s + (3.15 − 0.153i)10-s + (−2.91 + 0.946i)11-s + (−0.200 − 1.98i)12-s + (−1.26 − 2.47i)13-s + (−3.72 + 0.481i)14-s + (1.90 − 1.16i)15-s + (−3.48 + 1.96i)16-s + (−0.901 − 5.68i)17-s + ⋯
L(s)  = 1  + (−0.791 − 0.610i)2-s + (−0.570 − 0.0903i)3-s + (0.254 + 0.967i)4-s + (−0.761 + 0.648i)5-s + (0.396 + 0.419i)6-s + (0.708 − 0.708i)7-s + (0.388 − 0.921i)8-s + (0.317 + 0.103i)9-s + (0.998 − 0.0483i)10-s + (−0.878 + 0.285i)11-s + (−0.0577 − 0.574i)12-s + (−0.349 − 0.686i)13-s + (−0.994 + 0.128i)14-s + (0.492 − 0.300i)15-s + (−0.870 + 0.492i)16-s + (−0.218 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.950 + 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0441516 - 0.276467i\)
\(L(\frac12)\) \(\approx\) \(0.0441516 - 0.276467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.863i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (1.70 - 1.44i)T \)
good7 \( 1 + (-1.87 + 1.87i)T - 7iT^{2} \)
11 \( 1 + (2.91 - 0.946i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.26 + 2.47i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.901 + 5.68i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.66 + 2.66i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.07 - 6.04i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (4.48 + 6.17i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.14 + 1.57i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (4.87 - 2.48i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.46 - 4.51i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (0.964 + 0.964i)T + 43iT^{2} \)
47 \( 1 + (-1.87 + 11.8i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-1.98 + 12.5i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.89 + 5.81i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.51 - 7.73i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.31 - 0.208i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-7.46 - 10.2i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.04 + 1.55i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (0.348 - 0.252i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.82 - 11.5i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-2.37 + 0.771i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (12.2 + 1.93i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.42958805374791909035722593236, −10.44907692446912858683128854584, −9.829495981029466775335622308707, −8.248900802410921933431311793223, −7.55231899795309521048359492591, −6.89650137220395459757559216455, −5.05583462846607019930671144571, −3.84820089636604851112655879618, −2.38510295479986947214499875776, −0.27506758036635696547572948320, 1.84856032226710960199701378902, 4.34571310178408505070344498091, 5.31543464755530267066958368893, 6.26394575192446895936145386921, 7.56753192989616186912748629195, 8.414983021786728876673232501555, 8.975776837191003148258975462066, 10.46805997735451855062471128673, 10.96042852149869368979820085824, 12.09147370078149989725057523528

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