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Properties

Label 2-828-92.91-c1-0-35
Degree $2$
Conductor $828$
Sign $0.993 - 0.111i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
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  + (−1.11 + 0.872i)2-s + (0.477 − 1.94i)4-s − 0.970i·5-s + 4.31·7-s + (1.16 + 2.57i)8-s + (0.846 + 1.07i)10-s + 3.90·11-s + 1.84·13-s + (−4.80 + 3.76i)14-s + (−3.54 − 1.85i)16-s − 0.465i·17-s − 6.89·19-s + (−1.88 − 0.463i)20-s + (−4.34 + 3.40i)22-s + (−1.65 − 4.50i)23-s + ⋯
L(s)  = 1  + (−0.786 + 0.617i)2-s + (0.238 − 0.971i)4-s − 0.433i·5-s + 1.63·7-s + (0.411 + 0.911i)8-s + (0.267 + 0.341i)10-s + 1.17·11-s + 0.511·13-s + (−1.28 + 1.00i)14-s + (−0.886 − 0.463i)16-s − 0.112i·17-s − 1.58·19-s + (−0.421 − 0.103i)20-s + (−0.925 + 0.726i)22-s + (−0.345 − 0.938i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34992 + 0.0752489i\)
\(L(\frac12)\) \(\approx\) \(1.34992 + 0.0752489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.872i)T \)
3 \( 1 \)
23 \( 1 + (1.65 + 4.50i)T \)
good5 \( 1 + 0.970iT - 5T^{2} \)
7 \( 1 - 4.31T + 7T^{2} \)
11 \( 1 - 3.90T + 11T^{2} \)
13 \( 1 - 1.84T + 13T^{2} \)
17 \( 1 + 0.465iT - 17T^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 + 2.44iT - 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 5.37T + 43T^{2} \)
47 \( 1 + 8.65iT - 47T^{2} \)
53 \( 1 - 10.2iT - 53T^{2} \)
59 \( 1 + 8.30iT - 59T^{2} \)
61 \( 1 + 7.91iT - 61T^{2} \)
67 \( 1 - 2.57T + 67T^{2} \)
71 \( 1 - 6.53iT - 71T^{2} \)
73 \( 1 + 1.48T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 4.10iT - 89T^{2} \)
97 \( 1 + 5.32iT - 97T^{2} \)
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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

−10.20139358845313530224078592019, −9.011579775820985937723241776927, −8.549054649448875238633999361722, −7.951704949577847232174863462853, −6.80991145546285679724625885065, −6.06995481359429834920554121211, −4.89548234631967319450828771386, −4.25225012209184615089273949122, −2.09003681656168769646856462076, −1.09245457731062068928521042786, 1.31293597674881945814447065642, 2.23343174491869915342394760533, 3.74507112251372010490528924254, 4.48241237219863440851546543958, 5.97392487012365581890126701310, 7.03667992994631170377941534174, 7.82147558054999238256412404771, 8.691880154589441181874367276808, 9.177027543080106963592078138289, 10.47318631520165685374089614237

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