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Properties

Label 2-2217-2217.776-c0-0-0
Degree $2$
Conductor $2217$
Sign $-0.243 - 0.970i$
Analytic cond. $1.10642$
Root an. cond. $1.05186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.973 + 0.227i)3-s + (−0.997 + 0.0765i)4-s + (0.0551 + 1.43i)7-s + (0.896 − 0.443i)9-s + (0.953 − 0.301i)12-s + (0.590 − 0.912i)13-s + (0.988 − 0.152i)16-s + (−0.353 + 0.142i)19-s + (−0.381 − 1.38i)21-s + (0.896 + 0.443i)25-s + (−0.771 + 0.636i)27-s + (−0.165 − 1.43i)28-s + (−0.974 + 1.50i)31-s + (−0.859 + 0.511i)36-s + (0.668 − 0.103i)37-s + ⋯
L(s)  = 1  + (−0.973 + 0.227i)3-s + (−0.997 + 0.0765i)4-s + (0.0551 + 1.43i)7-s + (0.896 − 0.443i)9-s + (0.953 − 0.301i)12-s + (0.590 − 0.912i)13-s + (0.988 − 0.152i)16-s + (−0.353 + 0.142i)19-s + (−0.381 − 1.38i)21-s + (0.896 + 0.443i)25-s + (−0.771 + 0.636i)27-s + (−0.165 − 1.43i)28-s + (−0.974 + 1.50i)31-s + (−0.859 + 0.511i)36-s + (0.668 − 0.103i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $-0.243 - 0.970i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ -0.243 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5686914462\)
\(L(\frac12)\) \(\approx\) \(0.5686914462\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.973 - 0.227i)T \)
739 \( 1 + (-0.953 + 0.301i)T \)
good2 \( 1 + (0.997 - 0.0765i)T^{2} \)
5 \( 1 + (-0.896 - 0.443i)T^{2} \)
7 \( 1 + (-0.0551 - 1.43i)T + (-0.997 + 0.0765i)T^{2} \)
11 \( 1 + (0.665 + 0.746i)T^{2} \)
13 \( 1 + (-0.590 + 0.912i)T + (-0.409 - 0.912i)T^{2} \)
17 \( 1 + (-0.0383 - 0.999i)T^{2} \)
19 \( 1 + (0.353 - 0.142i)T + (0.720 - 0.693i)T^{2} \)
23 \( 1 + (0.997 - 0.0765i)T^{2} \)
29 \( 1 + (-0.477 + 0.878i)T^{2} \)
31 \( 1 + (0.974 - 1.50i)T + (-0.409 - 0.912i)T^{2} \)
37 \( 1 + (-0.668 + 0.103i)T + (0.953 - 0.301i)T^{2} \)
41 \( 1 + (0.543 + 0.839i)T^{2} \)
43 \( 1 + (0.293 + 0.242i)T + (0.190 + 0.981i)T^{2} \)
47 \( 1 + (0.114 + 0.993i)T^{2} \)
53 \( 1 + (-0.953 + 0.301i)T^{2} \)
59 \( 1 + (-0.190 + 0.981i)T^{2} \)
61 \( 1 + (0.523 - 1.90i)T + (-0.859 - 0.511i)T^{2} \)
67 \( 1 + (1.78 - 0.884i)T + (0.606 - 0.795i)T^{2} \)
71 \( 1 + (0.114 - 0.993i)T^{2} \)
73 \( 1 + (-1.08 - 1.42i)T + (-0.264 + 0.964i)T^{2} \)
79 \( 1 + (-0.0258 + 0.0720i)T + (-0.771 - 0.636i)T^{2} \)
83 \( 1 + (0.771 + 0.636i)T^{2} \)
89 \( 1 + (0.409 - 0.912i)T^{2} \)
97 \( 1 + (-0.0730 - 1.90i)T + (-0.997 + 0.0765i)T^{2} \)
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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

−9.277189806815797920913843227633, −8.858643204531286971564719503535, −8.118455855123919663721681627471, −7.03911513681582708107522538530, −5.97731752567425585878392802095, −5.48531045309748371378356626763, −4.91004548334057812221481443547, −3.88797752154972848071670944971, −2.90913641293950434325645301838, −1.28238684959737203619566160999, 0.54045858159703644186672873750, 1.67293755292336268896683592853, 3.58751914896090267635537069061, 4.37281468322826498099642526924, 4.78163330454442602752210053679, 5.94282677661184337130626131275, 6.59054413536266735395599978690, 7.44564100201693139754230370301, 8.103361327594246131917552621942, 9.178173398387000961472975476440

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