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Properties

Label 2-2627-2627.141-c0-0-3
Degree $2$
Conductor $2627$
Sign $-0.978 - 0.206i$
Analytic cond. $1.31104$
Root an. cond. $1.14500$
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  + (0.534 + 1.46i)2-s + (1.87 + 0.680i)3-s + (−1.10 + 0.928i)4-s + (−1.18 + 0.209i)5-s + 3.11i·6-s + (−0.602 − 0.347i)8-s + (2.26 + 1.90i)9-s + (−0.944 − 1.63i)10-s + (−2.70 + 0.983i)12-s + (−2.36 − 0.417i)15-s + (−0.0619 + 0.351i)16-s + (−1.58 + 4.34i)18-s + (0.683 − 1.87i)19-s + (1.12 − 1.33i)20-s + (−0.890 − 1.06i)24-s + (0.430 − 0.156i)25-s + ⋯
L(s)  = 1  + (0.534 + 1.46i)2-s + (1.87 + 0.680i)3-s + (−1.10 + 0.928i)4-s + (−1.18 + 0.209i)5-s + 3.11i·6-s + (−0.602 − 0.347i)8-s + (2.26 + 1.90i)9-s + (−0.944 − 1.63i)10-s + (−2.70 + 0.983i)12-s + (−2.36 − 0.417i)15-s + (−0.0619 + 0.351i)16-s + (−1.58 + 4.34i)18-s + (0.683 − 1.87i)19-s + (1.12 − 1.33i)20-s + (−0.890 − 1.06i)24-s + (0.430 − 0.156i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2627\)    =    \(37 \cdot 71\)
Sign: $-0.978 - 0.206i$
Analytic conductor: \(1.31104\)
Root analytic conductor: \(1.14500\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2627} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2627,\ (\ :0),\ -0.978 - 0.206i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.476733029\)
\(L(\frac12)\) \(\approx\) \(2.476733029\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-0.0249 + 0.999i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
good2 \( 1 + (-0.534 - 1.46i)T + (-0.766 + 0.642i)T^{2} \)
3 \( 1 + (-1.87 - 0.680i)T + (0.766 + 0.642i)T^{2} \)
5 \( 1 + (1.18 - 0.209i)T + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.683 + 1.87i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.72 + 0.997i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 - 0.0996iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 - 1.84T + T^{2} \)
79 \( 1 + (-0.765 + 0.134i)T + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (1.51 + 1.27i)T + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (-1.79 - 0.316i)T + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.142305891147003477808842896427, −8.386471933779539393529777531660, −7.67468091479276382673785388700, −7.47920617676979462962973590364, −6.70404433419758364642058123928, −5.30146045310262336698507205111, −4.57572895328325068168043832990, −3.92045346204429067107692768518, −3.32851379521701251292151362903, −2.24343706800262459641354427499, 1.26340663878183122499238588721, 1.99037363566179431489000368178, 3.06462009984600912375412650385, 3.71458230411583986391212213772, 3.93320790748062760946358326373, 5.13921649212365062048730050499, 6.64542553144003265044622815729, 7.66445199317497157124777982697, 7.88141270789438696042816002699, 8.742644473717276137787261566287

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