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Properties

Label 2-737-737.351-c0-0-0
Degree $2$
Conductor $737$
Sign $0.454 + 0.890i$
Analytic cond. $0.367810$
Root an. cond. $0.606474$
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  + (−0.544 − 1.19i)3-s + (0.928 + 0.371i)4-s + (−0.0913 + 0.0268i)5-s + (−0.468 + 0.540i)9-s + (0.235 − 0.971i)11-s + (−0.0623 − 1.30i)12-s + (0.0816 + 0.0941i)15-s + (0.723 + 0.690i)16-s + (−0.0947 − 0.00904i)20-s + (0.481 − 0.676i)23-s + (−0.833 + 0.535i)25-s + (−0.357 − 0.105i)27-s + (0.581 − 0.299i)31-s + (−1.28 + 0.247i)33-s + (−0.635 + 0.327i)36-s + (0.327 + 0.566i)37-s + ⋯
L(s)  = 1  + (−0.544 − 1.19i)3-s + (0.928 + 0.371i)4-s + (−0.0913 + 0.0268i)5-s + (−0.468 + 0.540i)9-s + (0.235 − 0.971i)11-s + (−0.0623 − 1.30i)12-s + (0.0816 + 0.0941i)15-s + (0.723 + 0.690i)16-s + (−0.0947 − 0.00904i)20-s + (0.481 − 0.676i)23-s + (−0.833 + 0.535i)25-s + (−0.357 − 0.105i)27-s + (0.581 − 0.299i)31-s + (−1.28 + 0.247i)33-s + (−0.635 + 0.327i)36-s + (0.327 + 0.566i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ 0.454 + 0.890i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9823678878\)
\(L(\frac12)\) \(\approx\) \(0.9823678878\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
good2 \( 1 + (-0.928 - 0.371i)T^{2} \)
3 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.0913 - 0.0268i)T + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.786 - 0.618i)T^{2} \)
13 \( 1 + (-0.580 - 0.814i)T^{2} \)
17 \( 1 + (-0.723 + 0.690i)T^{2} \)
19 \( 1 + (0.786 + 0.618i)T^{2} \)
23 \( 1 + (-0.481 + 0.676i)T + (-0.327 - 0.945i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.581 + 0.299i)T + (0.580 - 0.814i)T^{2} \)
37 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (1.95 + 0.186i)T + (0.981 + 0.189i)T^{2} \)
53 \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.396 - 0.254i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.888 - 0.458i)T^{2} \)
71 \( 1 + (-1.56 - 0.625i)T + (0.723 + 0.690i)T^{2} \)
73 \( 1 + (0.888 - 0.458i)T^{2} \)
79 \( 1 + (0.995 - 0.0950i)T^{2} \)
83 \( 1 + (-0.0475 - 0.998i)T^{2} \)
89 \( 1 + (0.738 - 1.61i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.80984170413097547046601542096, −9.635278576107895198939185976135, −8.292668703412235535617862844531, −7.81255113088588178259808108418, −6.69563152567766572790447732092, −6.41588656561212148481949100205, −5.37591932014226898205869492904, −3.72994676618597513089481595673, −2.56707178261188949432709710975, −1.29279331669749068583501158313, 1.88235187336754045450457269563, 3.35064095550687568269257423549, 4.49104151456441491017116237019, 5.27612898039559191653060412696, 6.23640732068394546012211088591, 7.12365376138732040791346841077, 8.104382403482254590001208401821, 9.580841209620206995853548058096, 9.834439975986727054027697177472, 10.74305884371968549052729737936

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