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Properties

Label 2-2217-2217.293-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.0949 + 0.995i$
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  + (−0.264 − 0.964i)3-s + (0.817 − 0.575i)4-s + (1.88 − 0.596i)7-s + (−0.859 + 0.511i)9-s + (−0.771 − 0.636i)12-s + (0.0263 + 0.227i)13-s + (0.338 − 0.941i)16-s + (−0.0763 − 0.00586i)19-s + (−1.07 − 1.65i)21-s + (−0.859 − 0.511i)25-s + (0.720 + 0.693i)27-s + (1.19 − 1.57i)28-s + (0.197 + 1.70i)31-s + (−0.409 + 0.912i)36-s + (−0.627 + 1.74i)37-s + ⋯
L(s)  = 1  + (−0.264 − 0.964i)3-s + (0.817 − 0.575i)4-s + (1.88 − 0.596i)7-s + (−0.859 + 0.511i)9-s + (−0.771 − 0.636i)12-s + (0.0263 + 0.227i)13-s + (0.338 − 0.941i)16-s + (−0.0763 − 0.00586i)19-s + (−1.07 − 1.65i)21-s + (−0.859 − 0.511i)25-s + (0.720 + 0.693i)27-s + (1.19 − 1.57i)28-s + (0.197 + 1.70i)31-s + (−0.409 + 0.912i)36-s + (−0.627 + 1.74i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0949 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0949 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.0949 + 0.995i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.0949 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552076283\)
\(L(\frac12)\) \(\approx\) \(1.552076283\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.264 + 0.964i)T \)
739 \( 1 + (0.771 + 0.636i)T \)
good2 \( 1 + (-0.817 + 0.575i)T^{2} \)
5 \( 1 + (0.859 + 0.511i)T^{2} \)
7 \( 1 + (-1.88 + 0.596i)T + (0.817 - 0.575i)T^{2} \)
11 \( 1 + (-0.896 - 0.443i)T^{2} \)
13 \( 1 + (-0.0263 - 0.227i)T + (-0.973 + 0.227i)T^{2} \)
17 \( 1 + (-0.953 + 0.301i)T^{2} \)
19 \( 1 + (0.0763 + 0.00586i)T + (0.988 + 0.152i)T^{2} \)
23 \( 1 + (-0.817 + 0.575i)T^{2} \)
29 \( 1 + (0.665 + 0.746i)T^{2} \)
31 \( 1 + (-0.197 - 1.70i)T + (-0.973 + 0.227i)T^{2} \)
37 \( 1 + (0.627 - 1.74i)T + (-0.771 - 0.636i)T^{2} \)
41 \( 1 + (0.114 - 0.993i)T^{2} \)
43 \( 1 + (-0.0551 + 0.0531i)T + (0.0383 - 0.999i)T^{2} \)
47 \( 1 + (-0.606 + 0.795i)T^{2} \)
53 \( 1 + (0.771 + 0.636i)T^{2} \)
59 \( 1 + (-0.0383 - 0.999i)T^{2} \)
61 \( 1 + (0.367 - 0.567i)T + (-0.409 - 0.912i)T^{2} \)
67 \( 1 + (1.40 - 0.835i)T + (0.477 - 0.878i)T^{2} \)
71 \( 1 + (-0.606 - 0.795i)T^{2} \)
73 \( 1 + (0.821 + 1.51i)T + (-0.543 + 0.839i)T^{2} \)
79 \( 1 + (1.76 - 0.712i)T + (0.720 - 0.693i)T^{2} \)
83 \( 1 + (-0.720 + 0.693i)T^{2} \)
89 \( 1 + (0.973 + 0.227i)T^{2} \)
97 \( 1 + (1.47 - 0.465i)T + (0.817 - 0.575i)T^{2} \)
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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

−8.708128003012468466571196038620, −8.147840141967325216095099843395, −7.37991659751747203544179191198, −6.88451253370373101636964857729, −5.96699815108075007090485455151, −5.18392868002054872542855398156, −4.46790832626686694338463081716, −2.89792499735619542919935010695, −1.75313675511752624039962532020, −1.31251587677796509501716423805, 1.77959498243479330931543280387, 2.65566052207856333344340421117, 3.84808874325487257194508740048, 4.52930180884995823798431607659, 5.55006562997588126468069942422, 5.95943989328872976312197758637, 7.32726818976710274114761181203, 7.920643839799205919491933391822, 8.594523480884514909268942472062, 9.309724948327870448768790597343

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