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Properties

Label 2-294-21.20-c5-0-12
Degree $2$
Conductor $294$
Sign $-0.373 - 0.927i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + (−10.8 + 11.2i)3-s − 16·4-s − 104.·5-s + (−44.9 − 43.2i)6-s − 64i·8-s + (−9.33 − 242. i)9-s − 419. i·10-s − 83.8i·11-s + (172. − 179. i)12-s + 382. i·13-s + (1.13e3 − 1.17e3i)15-s + 256·16-s − 1.43e3·17-s + (971. − 37.3i)18-s − 2.24e3i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.693 + 0.720i)3-s − 0.5·4-s − 1.87·5-s + (−0.509 − 0.490i)6-s − 0.353i·8-s + (−0.0384 − 0.999i)9-s − 1.32i·10-s − 0.208i·11-s + (0.346 − 0.360i)12-s + 0.628i·13-s + (1.29 − 1.35i)15-s + 0.250·16-s − 1.20·17-s + (0.706 − 0.0271i)18-s − 1.42i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ -0.373 - 0.927i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3439256839\)
\(L(\frac12)\) \(\approx\) \(0.3439256839\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
3 \( 1 + (10.8 - 11.2i)T \)
7 \( 1 \)
good5 \( 1 + 104.T + 3.12e3T^{2} \)
11 \( 1 + 83.8iT - 1.61e5T^{2} \)
13 \( 1 - 382. iT - 3.71e5T^{2} \)
17 \( 1 + 1.43e3T + 1.41e6T^{2} \)
19 \( 1 + 2.24e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.69e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.17e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.36e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.14e4T + 6.93e7T^{2} \)
41 \( 1 + 3.03e3T + 1.15e8T^{2} \)
43 \( 1 + 1.28e4T + 1.47e8T^{2} \)
47 \( 1 + 4.80e3T + 2.29e8T^{2} \)
53 \( 1 - 858. iT - 4.18e8T^{2} \)
59 \( 1 + 3.19e4T + 7.14e8T^{2} \)
61 \( 1 - 2.90e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.70e4T + 1.35e9T^{2} \)
71 \( 1 + 3.39e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.24e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.06e4T + 3.07e9T^{2} \)
83 \( 1 + 1.58e4T + 3.93e9T^{2} \)
89 \( 1 + 1.44e5T + 5.58e9T^{2} \)
97 \( 1 + 7.65e4iT - 8.58e9T^{2} \)
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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

−11.20545867458418565883333107380, −10.53367108472170535745678608379, −8.924620524802212093858793500480, −8.548228097463028054744212014163, −7.06287602753348041771750186843, −6.59678603115269440921519170358, −4.82545440493059264371289882214, −4.44571681934408645338207778002, −3.28127376128120027420217036433, −0.44720427278187012541325096198, 0.25856805584792329759853516258, 1.64345601480952341771490549808, 3.31738712515443015350632592539, 4.30359678062316363772355705700, 5.46249849096673015204109100532, 6.92355319503798119145526761319, 7.82443331517056686957039510968, 8.434018673047925058057673732121, 10.00628463283514185154544758066, 11.10624525611457227411542643411

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