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Properties

Label 2-300-100.27-c1-0-21
Degree $2$
Conductor $300$
Sign $0.916 - 0.399i$
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  + (1.17 + 0.786i)2-s + (0.891 − 0.453i)3-s + (0.761 + 1.84i)4-s + (−0.0577 − 2.23i)5-s + (1.40 + 0.167i)6-s + (0.0211 − 0.0211i)7-s + (−0.560 + 2.77i)8-s + (0.587 − 0.809i)9-s + (1.69 − 2.67i)10-s + (1.52 + 2.10i)11-s + (1.51 + 1.30i)12-s + (3.59 − 0.569i)13-s + (0.0415 − 0.00822i)14-s + (−1.06 − 1.96i)15-s + (−2.84 + 2.81i)16-s + (−1.43 + 2.81i)17-s + ⋯
L(s)  = 1  + (0.830 + 0.556i)2-s + (0.514 − 0.262i)3-s + (0.380 + 0.924i)4-s + (−0.0258 − 0.999i)5-s + (0.573 + 0.0684i)6-s + (0.00800 − 0.00800i)7-s + (−0.198 + 0.980i)8-s + (0.195 − 0.269i)9-s + (0.534 − 0.844i)10-s + (0.460 + 0.633i)11-s + (0.438 + 0.375i)12-s + (0.996 − 0.157i)13-s + (0.0111 − 0.00219i)14-s + (−0.275 − 0.507i)15-s + (−0.710 + 0.704i)16-s + (−0.347 + 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.916 - 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29705 + 0.478584i\)
\(L(\frac12)\) \(\approx\) \(2.29705 + 0.478584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.786i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
5 \( 1 + (0.0577 + 2.23i)T \)
good7 \( 1 + (-0.0211 + 0.0211i)T - 7iT^{2} \)
11 \( 1 + (-1.52 - 2.10i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.59 + 0.569i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.43 - 2.81i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.41 + 4.35i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.82 + 0.922i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (7.62 + 2.47i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.76 + 0.573i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.30 - 8.25i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.210 + 0.153i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \)
47 \( 1 + (5.36 + 10.5i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (1.61 + 3.17i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-0.594 - 0.432i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.96 + 2.88i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.97 - 1.51i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-10.8 - 3.51i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.68 - 10.6i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-4.55 + 14.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (5.86 - 11.5i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (5.58 + 7.68i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.72 + 2.91i)T + (57.0 - 78.4i)T^{2} \)
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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

−12.08423275386305034400266358801, −11.24324739137299683154784720397, −9.676468506125361028622848746841, −8.596203018651161086405690726192, −8.056008162290649773917248211911, −6.80962102045974202746422620581, −5.86425896285261902886992886289, −4.56661221487483310570572444184, −3.73633647282102522967201829031, −1.96792795876331655168361048069, 2.00630804180399149022665292345, 3.38993116003362865963062725478, 4.02881038648275742794896578691, 5.70928673391969464174994490835, 6.51373726232772237624924423399, 7.74684504422843957587719294657, 9.095702259371871837850885239977, 10.02754152264229400808413889679, 10.97302514275192249318850969438, 11.47405030044294200306656383659

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