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Properties

Label 2-644-92.91-c1-0-44
Degree $2$
Conductor $644$
Sign $0.999 - 0.00432i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (0.425 + 1.34i)2-s − 1.30i·3-s + (−1.63 + 1.14i)4-s − 0.616i·5-s + (1.76 − 0.556i)6-s − 7-s + (−2.24 − 1.72i)8-s + 1.28·9-s + (0.832 − 0.262i)10-s + 1.66·11-s + (1.50 + 2.14i)12-s + 3.32·13-s + (−0.425 − 1.34i)14-s − 0.807·15-s + (1.37 − 3.75i)16-s − 7.61i·17-s + ⋯
L(s)  = 1  + (0.300 + 0.953i)2-s − 0.756i·3-s + (−0.819 + 0.573i)4-s − 0.275i·5-s + (0.721 − 0.227i)6-s − 0.377·7-s + (−0.793 − 0.609i)8-s + 0.428·9-s + (0.263 − 0.0829i)10-s + 0.501·11-s + (0.433 + 0.619i)12-s + 0.923·13-s + (−0.113 − 0.360i)14-s − 0.208·15-s + (0.342 − 0.939i)16-s − 1.84i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.999 - 0.00432i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.999 - 0.00432i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61811 + 0.00349712i\)
\(L(\frac12)\) \(\approx\) \(1.61811 + 0.00349712i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.425 - 1.34i)T \)
7 \( 1 + T \)
23 \( 1 + (-3.94 - 2.73i)T \)
good3 \( 1 + 1.30iT - 3T^{2} \)
5 \( 1 + 0.616iT - 5T^{2} \)
11 \( 1 - 1.66T + 11T^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
17 \( 1 + 7.61iT - 17T^{2} \)
19 \( 1 + 5.00T + 19T^{2} \)
29 \( 1 - 8.34T + 29T^{2} \)
31 \( 1 - 1.61iT - 31T^{2} \)
37 \( 1 + 1.52iT - 37T^{2} \)
41 \( 1 + 5.75T + 41T^{2} \)
43 \( 1 - 6.32T + 43T^{2} \)
47 \( 1 - 0.313iT - 47T^{2} \)
53 \( 1 + 1.93iT - 53T^{2} \)
59 \( 1 - 0.633iT - 59T^{2} \)
61 \( 1 + 2.00iT - 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 5.31iT - 71T^{2} \)
73 \( 1 + 5.29T + 73T^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 - 8.64T + 83T^{2} \)
89 \( 1 - 15.3iT - 89T^{2} \)
97 \( 1 - 2.19iT - 97T^{2} \)
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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

−10.48180905753549851240910383800, −9.293461154074811243741169235350, −8.747887314887340818196373174079, −7.72317344520636464620246428509, −6.80911703505124208138953529967, −6.44783174139684946864210266786, −5.15456348921097990894799563185, −4.24028466926658017008502941303, −2.93991211692854349283446231511, −0.959198960668592594642240824590, 1.43538409552439971106362594645, 2.99440395767584278914561961339, 3.98909551523423613729594372751, 4.56288045565888961564471285900, 5.97059329412962473043242106219, 6.69585123835354282325009991112, 8.491525622874537213241913081310, 8.909622363891764903828707656890, 10.10709726073039341347393385722, 10.54117078089182198565419767780

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