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Properties

Label 2-294-7.6-c4-0-25
Degree $2$
Conductor $294$
Sign $-0.409 + 0.912i$
Analytic cond. $30.3907$
Root an. cond. $5.51278$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s + 5.19i·3-s + 8.00·4-s − 41.5i·5-s + 14.6i·6-s + 22.6·8-s − 27·9-s − 117. i·10-s − 50.8·11-s + 41.5i·12-s + 166. i·13-s + 216.·15-s + 64.0·16-s − 538. i·17-s − 76.3·18-s − 398. i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s − 1.66i·5-s + 0.408i·6-s + 0.353·8-s − 0.333·9-s − 1.17i·10-s − 0.419·11-s + 0.288i·12-s + 0.987i·13-s + 0.960·15-s + 0.250·16-s − 1.86i·17-s − 0.235·18-s − 1.10i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(30.3907\)
Root analytic conductor: \(5.51278\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :2),\ -0.409 + 0.912i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.010240969\)
\(L(\frac12)\) \(\approx\) \(2.010240969\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
3 \( 1 - 5.19iT \)
7 \( 1 \)
good5 \( 1 + 41.5iT - 625T^{2} \)
11 \( 1 + 50.8T + 1.46e4T^{2} \)
13 \( 1 - 166. iT - 2.85e4T^{2} \)
17 \( 1 + 538. iT - 8.35e4T^{2} \)
19 \( 1 + 398. iT - 1.30e5T^{2} \)
23 \( 1 + 573.T + 2.79e5T^{2} \)
29 \( 1 + 831.T + 7.07e5T^{2} \)
31 \( 1 - 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 - 918.T + 1.87e6T^{2} \)
41 \( 1 + 1.58e3iT - 2.82e6T^{2} \)
43 \( 1 + 377.T + 3.41e6T^{2} \)
47 \( 1 + 3.70e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.32e3T + 7.89e6T^{2} \)
59 \( 1 + 6.26e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.78e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.12e3T + 2.01e7T^{2} \)
71 \( 1 - 1.66e3T + 2.54e7T^{2} \)
73 \( 1 - 3.97e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.18e4T + 3.89e7T^{2} \)
83 \( 1 - 5.62e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.25e4iT - 6.27e7T^{2} \)
97 \( 1 + 4.81e3iT - 8.85e7T^{2} \)
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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

−11.13644401507413442527190887135, −9.647815298977652362346741162284, −9.131259070735898567849509810351, −8.030082577456660128985758160513, −6.75653995220719198186899459130, −5.19609416404075153355484471698, −4.92931077121018528994569358674, −3.78684433718142227595558232601, −2.14867163209406534921746745980, −0.45207884228797123321381168639, 1.87756941185106490660386562332, 2.99302650979187050369648677915, 3.95467191901046689643885712597, 5.99591559460151289460474278512, 6.10851210740583176946371828729, 7.60060774124652574261455218581, 7.985351999120948150913953153569, 9.991652115242576793987249448306, 10.64617467111811370818031542669, 11.40086115146116001415296684015

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