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Properties

Label 2-294-21.20-c1-0-10
Degree $2$
Conductor $294$
Sign $-0.394 + 0.918i$
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  − i·2-s + (0.923 − 1.46i)3-s − 4-s + 1.53·5-s + (−1.46 − 0.923i)6-s + i·8-s + (−1.29 − 2.70i)9-s − 1.53i·10-s − 6.24i·11-s + (−0.923 + 1.46i)12-s + 5.22i·13-s + (1.41 − 2.24i)15-s + 16-s + 2.29·17-s + (−2.70 + 1.29i)18-s − 1.84i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.533 − 0.845i)3-s − 0.5·4-s + 0.684·5-s + (−0.598 − 0.377i)6-s + 0.353i·8-s + (−0.430 − 0.902i)9-s − 0.484i·10-s − 1.88i·11-s + (−0.266 + 0.422i)12-s + 1.44i·13-s + (0.365 − 0.579i)15-s + 0.250·16-s + 0.556·17-s + (−0.638 + 0.304i)18-s − 0.423i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.394 + 0.918i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855656 - 1.29837i\)
\(L(\frac12)\) \(\approx\) \(0.855656 - 1.29837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.923 + 1.46i)T \)
7 \( 1 \)
good5 \( 1 - 1.53T + 5T^{2} \)
11 \( 1 + 6.24iT - 11T^{2} \)
13 \( 1 - 5.22iT - 13T^{2} \)
17 \( 1 - 2.29T + 17T^{2} \)
19 \( 1 + 1.84iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 - 6.75iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 9.23T + 41T^{2} \)
43 \( 1 - 5.07T + 43T^{2} \)
47 \( 1 - 3.69T + 47T^{2} \)
53 \( 1 - 6.48iT - 53T^{2} \)
59 \( 1 + 4.46T + 59T^{2} \)
61 \( 1 - 5.86iT - 61T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 + 5.17iT - 71T^{2} \)
73 \( 1 - 0.765iT - 73T^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 + 0.131T + 83T^{2} \)
89 \( 1 - 6.17T + 89T^{2} \)
97 \( 1 - 2.48iT - 97T^{2} \)
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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

−11.56408007918524415089924240455, −10.72525440616264665275958562421, −9.311294551163053533953823340953, −8.921761777249990690067783932424, −7.77336003650304662127841201801, −6.48203995536721822544266904164, −5.58355464166606377110210991656, −3.78082401371752756292163506518, −2.63695045269620370516864311154, −1.27676627917312319132880580012, 2.37395390589446238507368970073, 3.97871034646898278512024523765, 5.06742174350618448282538064581, 5.94994373661983316460935237404, 7.45875818486743399805476124146, 8.124030583205321513786416502369, 9.490441124171410986024356624992, 9.859917155288938934254637576309, 10.70867656540530054406642759636, 12.35339974733241332279597756667

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