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Properties

Label 1-1792-1792.107-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0878 - 0.996i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.935 − 0.352i)3-s + (0.683 − 0.729i)5-s + (0.751 − 0.659i)9-s + (0.812 + 0.582i)11-s + (−0.956 − 0.290i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.528 − 0.849i)19-s + (0.997 + 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.0327 − 0.999i)37-s + ⋯
L(s)  = 1  + (0.935 − 0.352i)3-s + (0.683 − 0.729i)5-s + (0.751 − 0.659i)9-s + (0.812 + 0.582i)11-s + (−0.956 − 0.290i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.528 − 0.849i)19-s + (0.997 + 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.0327 − 0.999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0878 - 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0878 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.335048613 - 3.054008158i\)
\(L(\frac12)\) \(\approx\) \(3.335048613 - 3.054008158i\)
\(L(1)\) \(\approx\) \(1.767355699 - 0.6293611433i\)
\(L(1)\) \(\approx\) \(1.767355699 - 0.6293611433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.935 - 0.352i)T \)
5 \( 1 + (0.683 - 0.729i)T \)
11 \( 1 + (0.812 + 0.582i)T \)
13 \( 1 + (-0.956 - 0.290i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (0.528 - 0.849i)T \)
23 \( 1 + (0.997 + 0.0654i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (-0.0327 - 0.999i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (0.773 - 0.634i)T \)
47 \( 1 + (0.793 + 0.608i)T \)
53 \( 1 + (-0.582 + 0.812i)T \)
59 \( 1 + (-0.227 + 0.973i)T \)
61 \( 1 + (-0.986 - 0.162i)T \)
67 \( 1 + (0.352 + 0.935i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (-0.881 + 0.471i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.19989149208438730469217266419, −19.38116990169585777679499009676, −18.83222584624704392750457427739, −18.243703028853815415690509539060, −16.98591735540842179969608552067, −16.70856583265967100734167325932, −15.61344632517117246036633933094, −14.71907833108773358551497980369, −14.33327065465959569956551320996, −13.8445354390628041711889922343, −12.89706955764106464092570193029, −11.98269381769079528766046293620, −11.06964944351259621511084310261, −10.1524722733518467501359847905, −9.676895168589213718415279263608, −9.00064907720452258049559423676, −8.02526748882116437354783899117, −7.252909928897129901497115658053, −6.500768521183005769830225998810, −5.485715756499978083589294165662, −4.628906265353248953743924198332, −3.416060465146688156604630859862, −3.076630789353991353440132913922, −2.01012044829903933144522147813, −1.161455402368119281266611380598, 0.72011438328284812898060063286, 1.465193474903373914851545758735, 2.37801404850362043110611217057, 3.18515182583573627074981756148, 4.28503847608580171724265181020, 5.049600563683058276739577197167, 5.978181071266153470477196585181, 7.12852585123674233218448017139, 7.4815722879655185472069621698, 8.684576733372978246436624043262, 9.19146294501848945432099381651, 9.76307300158056200661422431826, 10.61251024261629414218546003706, 12.07730333714201279070819036017, 12.38754385142603266495519460569, 13.15941666145125299682206791937, 14.01763127975928490926734856380, 14.47155744298771995878320605232, 15.28421879693126434022110290353, 16.11684746441460913034411518765, 17.22312376720712605958934519924, 17.43512162486178413311566513393, 18.44352703904755167897332569257, 19.28207307609148531191899954451, 19.957397693763286865717533008

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