Properties

Label 1-736-736.229-r1-0-0
Degree $1$
Conductor $736$
Sign $0.831 - 0.555i$
Analytic cond. $79.0941$
Root an. cond. $79.0941$
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  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 15-s + 17-s + (−0.707 + 0.707i)19-s + (0.707 + 0.707i)21-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − i·7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 15-s + 17-s + (−0.707 + 0.707i)19-s + (0.707 + 0.707i)21-s + i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s + 31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(79.0941\)
Root analytic conductor: \(79.0941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 736,\ (1:\ ),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.312284742 - 0.3980772245i\)
\(L(\frac12)\) \(\approx\) \(1.312284742 - 0.3980772245i\)
\(L(1)\) \(\approx\) \(0.8428020512 - 0.03984645071i\)
\(L(1)\) \(\approx\) \(0.8428020512 - 0.03984645071i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - T \)
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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

−22.45007287320632742619535422300, −21.78519741321031868267687178249, −20.96411157526253091798676826245, −19.37344525613888962774921814739, −19.12093034442342785307047828121, −18.524522034033289047225935407236, −17.62376322262146110547297354179, −16.66180197629722443714927910719, −15.94340608485157962894731297328, −15.02486296546443942709476956354, −14.153580227640907009540386924683, −13.251842328171862595246474941366, −12.19126968341698155551872076371, −11.54881206053175242771651219337, −11.16586521759821944327632104957, −9.953624259312598552400158441, −8.65202584275913930042089349248, −8.05184966483422733394601578741, −6.81631297733821859044374444966, −6.336745624033343156118215121907, −5.42506328302431899108560017894, −4.17621170226680375871346088553, −3.07081013392137071628222806476, −1.988606315978428333380858404336, −0.73784145321136288541916162104, 0.55987225305125972816751243182, 1.38102937350266405968703198515, 3.546227177832568130779451620665, 3.96143877846668432059524524243, 4.88232803703133834875399296078, 5.81557789467737800783848885928, 6.9072938278079399593362501496, 7.8436819682160939481586255340, 8.82339546151914222076141733948, 9.840136428161112840239822639106, 10.552256881895771057255558153313, 11.369252061005619736308605601490, 12.30476230436575910931937307141, 12.83460946315251143586377771577, 14.173009379178507677785635184645, 14.99865286558270261164905829670, 15.85441048516729510828274153793, 16.59770880944461366924473362303, 17.11292762477775797272853733903, 17.9022825792606073390277366098, 19.13260663326844299829188183480, 19.998003244831648192574289644151, 20.73533555301577705601053662704, 21.13341554227993543432693600543, 22.609568269272952172302548595663

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