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Properties

Label 2-294-147.104-c1-0-19
Degree $2$
Conductor $294$
Sign $-0.712 + 0.701i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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  + (0.781 + 0.623i)2-s + (−0.423 − 1.67i)3-s + (0.222 + 0.974i)4-s + (−2.99 − 1.44i)5-s + (0.715 − 1.57i)6-s + (−2.37 − 1.16i)7-s + (−0.433 + 0.900i)8-s + (−2.64 + 1.42i)9-s + (−1.44 − 2.99i)10-s + (−1.67 − 1.33i)11-s + (1.54 − 0.786i)12-s + (0.710 + 0.566i)13-s + (−1.12 − 2.39i)14-s + (−1.15 + 5.64i)15-s + (−0.900 + 0.433i)16-s + (0.575 − 2.52i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (−0.244 − 0.969i)3-s + (0.111 + 0.487i)4-s + (−1.34 − 0.645i)5-s + (0.292 − 0.643i)6-s + (−0.897 − 0.440i)7-s + (−0.153 + 0.318i)8-s + (−0.880 + 0.474i)9-s + (−0.456 − 0.948i)10-s + (−0.506 − 0.403i)11-s + (0.445 − 0.227i)12-s + (0.197 + 0.157i)13-s + (−0.302 − 0.639i)14-s + (−0.298 + 1.45i)15-s + (−0.225 + 0.108i)16-s + (0.139 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.712 + 0.701i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.712 + 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253221 - 0.618549i\)
\(L(\frac12)\) \(\approx\) \(0.253221 - 0.618549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
3 \( 1 + (0.423 + 1.67i)T \)
7 \( 1 + (2.37 + 1.16i)T \)
good5 \( 1 + (2.99 + 1.44i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (1.67 + 1.33i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.710 - 0.566i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.575 + 2.52i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + 1.40iT - 19T^{2} \)
23 \( 1 + (-3.48 + 0.795i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.07 + 0.474i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 10.3iT - 31T^{2} \)
37 \( 1 + (-0.219 + 0.963i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + (-3.83 - 1.84i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (8.38 - 4.03i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-8.09 + 10.1i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.54 - 0.580i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (10.4 - 5.04i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (11.6 + 2.66i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + (-15.3 + 3.50i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.78 - 7.00i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 4.17T + 79T^{2} \)
83 \( 1 + (0.660 + 0.828i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.11 + 5.16i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−11.67613673701888453395500773417, −10.95073485053658452233336562672, −9.241136378571347564833605592277, −8.126378662617921720727232628754, −7.49094701846219009613321627689, −6.60701525423320957012076383111, −5.44957595063349483535023607765, −4.20718114053501680546107395102, −2.97500390601211424016738767170, −0.41509250611305989262689618377, 3.02363818554803930272718790755, 3.64444277016252807910951876745, 4.81180195671405414365185356864, 5.99342481358054300100544469517, 7.12931297291501286603585564649, 8.454691560682026379162628306967, 9.572502024899193245270906080089, 10.58854423670597641783649198822, 11.04745584492188436925227779239, 12.16520582654562861627480492299

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