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Properties

Label 2-3736-3736.163-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.698 - 0.715i$
Analytic cond. $1.86450$
Root an. cond. $1.36546$
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  + (−0.484 − 0.874i)2-s + (0.404 − 1.68i)3-s + (−0.530 + 0.847i)4-s + (−1.66 + 0.460i)6-s + (0.998 + 0.0539i)8-s + (−1.77 − 0.905i)9-s + (0.297 − 0.206i)11-s + (1.21 + 1.23i)12-s + (−0.436 − 0.899i)16-s + (−0.874 − 1.14i)17-s + (0.0671 + 1.99i)18-s + (−0.324 − 0.259i)19-s + (−0.324 − 0.160i)22-s + (0.494 − 1.65i)24-s + (0.813 − 0.581i)25-s + ⋯
L(s)  = 1  + (−0.484 − 0.874i)2-s + (0.404 − 1.68i)3-s + (−0.530 + 0.847i)4-s + (−1.66 + 0.460i)6-s + (0.998 + 0.0539i)8-s + (−1.77 − 0.905i)9-s + (0.297 − 0.206i)11-s + (1.21 + 1.23i)12-s + (−0.436 − 0.899i)16-s + (−0.874 − 1.14i)17-s + (0.0671 + 1.99i)18-s + (−0.324 − 0.259i)19-s + (−0.324 − 0.160i)22-s + (0.494 − 1.65i)24-s + (0.813 − 0.581i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.698 - 0.715i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8255364859\)
\(L(\frac12)\) \(\approx\) \(0.8255364859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.484 + 0.874i)T \)
467 \( 1 + (-0.399 + 0.916i)T \)
good3 \( 1 + (-0.404 + 1.68i)T + (-0.890 - 0.454i)T^{2} \)
5 \( 1 + (-0.813 + 0.581i)T^{2} \)
7 \( 1 + (-0.424 - 0.905i)T^{2} \)
11 \( 1 + (-0.297 + 0.206i)T + (0.349 - 0.936i)T^{2} \)
13 \( 1 + (-0.472 - 0.881i)T^{2} \)
17 \( 1 + (0.874 + 1.14i)T + (-0.259 + 0.965i)T^{2} \)
19 \( 1 + (0.324 + 0.259i)T + (0.220 + 0.975i)T^{2} \)
23 \( 1 + (0.805 + 0.592i)T^{2} \)
29 \( 1 + (-0.194 - 0.980i)T^{2} \)
31 \( 1 + (-0.650 + 0.759i)T^{2} \)
37 \( 1 + (0.575 - 0.817i)T^{2} \)
41 \( 1 + (-0.191 + 1.22i)T + (-0.952 - 0.305i)T^{2} \)
43 \( 1 + (1.37 + 0.481i)T + (0.781 + 0.624i)T^{2} \)
47 \( 1 + (0.507 - 0.861i)T^{2} \)
53 \( 1 + (-0.843 - 0.536i)T^{2} \)
59 \( 1 + (1.05 - 0.402i)T + (0.746 - 0.665i)T^{2} \)
61 \( 1 + (0.0740 + 0.997i)T^{2} \)
67 \( 1 + (-0.636 - 0.346i)T + (0.542 + 0.840i)T^{2} \)
71 \( 1 + (0.660 + 0.750i)T^{2} \)
73 \( 1 + (0.402 + 0.571i)T + (-0.337 + 0.941i)T^{2} \)
79 \( 1 + (0.100 - 0.994i)T^{2} \)
83 \( 1 + (1.13 - 0.185i)T + (0.948 - 0.317i)T^{2} \)
89 \( 1 + (-0.0290 - 0.0281i)T + (0.0337 + 0.999i)T^{2} \)
97 \( 1 + (0.541 - 1.90i)T + (-0.851 - 0.525i)T^{2} \)
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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

−8.381458986699057553550155165537, −7.52145197198952007957561122204, −6.99292209995038942257181476455, −6.37468934468935019736453002269, −5.17699770070560182445048857607, −4.16045145430365091032647993603, −3.01348956423095378563907547298, −2.46016217666022252394087521752, −1.58487957854534988430475140640, −0.52804404486284074462173936149, 1.75879926644617898887326798363, 3.13877178176383525817684258163, 4.07972890021522839197862886947, 4.59826851126675387255783556110, 5.32960212646022490130456639356, 6.18520810210292377757600793968, 6.89343267885813186371672333057, 8.001376295368495409468883211775, 8.571257576084134316624986221285, 9.032042099317531471076692859776

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