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Properties

Label 2-294-49.25-c1-0-3
Degree $2$
Conductor $294$
Sign $0.905 - 0.424i$
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  + (0.955 − 0.294i)2-s + (0.365 + 0.930i)3-s + (0.826 − 0.563i)4-s + (−0.0494 − 0.00745i)5-s + (0.623 + 0.781i)6-s + (−0.154 + 2.64i)7-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (−0.0494 + 0.00745i)10-s + (4.04 + 3.75i)11-s + (0.826 + 0.563i)12-s + (0.842 − 3.69i)13-s + (0.630 + 2.56i)14-s + (−0.0111 − 0.0487i)15-s + (0.365 − 0.930i)16-s + (0.205 − 2.74i)17-s + ⋯
L(s)  = 1  + (0.675 − 0.208i)2-s + (0.210 + 0.537i)3-s + (0.413 − 0.281i)4-s + (−0.0221 − 0.00333i)5-s + (0.254 + 0.319i)6-s + (−0.0584 + 0.998i)7-s + (0.220 − 0.276i)8-s + (−0.244 + 0.226i)9-s + (−0.0156 + 0.00235i)10-s + (1.21 + 1.13i)11-s + (0.238 + 0.162i)12-s + (0.233 − 1.02i)13-s + (0.168 + 0.686i)14-s + (−0.00287 − 0.0125i)15-s + (0.0913 − 0.232i)16-s + (0.0498 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97890 + 0.441132i\)
\(L(\frac12)\) \(\approx\) \(1.97890 + 0.441132i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.955 + 0.294i)T \)
3 \( 1 + (-0.365 - 0.930i)T \)
7 \( 1 + (0.154 - 2.64i)T \)
good5 \( 1 + (0.0494 + 0.00745i)T + (4.77 + 1.47i)T^{2} \)
11 \( 1 + (-4.04 - 3.75i)T + (0.822 + 10.9i)T^{2} \)
13 \( 1 + (-0.842 + 3.69i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.205 + 2.74i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (1.90 + 3.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.153 - 2.04i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (6.44 - 3.10i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-2.67 + 4.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.71 - 4.57i)T + (13.5 + 34.4i)T^{2} \)
41 \( 1 + (-2.65 + 3.32i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (6.70 + 8.41i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (9.99 - 3.08i)T + (38.8 - 26.4i)T^{2} \)
53 \( 1 + (0.152 - 0.103i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (-5.54 + 0.836i)T + (56.3 - 17.3i)T^{2} \)
61 \( 1 + (5.33 + 3.64i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (-5.31 + 9.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.1 - 4.86i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (8.28 + 2.55i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (-1.81 - 3.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.41 - 6.20i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (7.99 - 7.41i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 - 11.3T + 97T^{2} \)
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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

−11.83696950055579520375538699605, −11.16352274288881869900078435281, −9.815414396493176573908539108339, −9.312594669155288700381247690107, −8.044601379242752468300218537308, −6.74540165106210853248659613176, −5.63659427917342423273155198232, −4.64745151412450553356131366014, −3.47490379150748789675529371132, −2.15947323121549243925230264880, 1.57348925315741778688581094426, 3.52321938147805303208316033149, 4.24351468144298522666796300671, 6.05350924624391158435165131502, 6.57673938446006918882144432793, 7.72609503099532743939725795131, 8.636815520731450171033990017446, 9.836964582891504420161219804160, 11.19958052237089642813260873834, 11.64353869546849785484735493092

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