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Properties

Label 1-448-448.317-r0-0-0
Degree $1$
Conductor $448$
Sign $0.225 - 0.974i$
Analytic cond. $2.08050$
Root an. cond. $2.08050$
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  + (0.793 − 0.608i)3-s + (0.608 − 0.793i)5-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (0.382 − 0.923i)13-s − i·15-s + (−0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (−0.258 + 0.965i)23-s + (−0.258 − 0.965i)25-s + (−0.382 − 0.923i)27-s + (0.923 + 0.382i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)3-s + (0.608 − 0.793i)5-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (0.382 − 0.923i)13-s − i·15-s + (−0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (−0.258 + 0.965i)23-s + (−0.258 − 0.965i)25-s + (−0.382 − 0.923i)27-s + (0.923 + 0.382i)29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.608 + 0.793i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.225 - 0.974i$
Analytic conductor: \(2.08050\)
Root analytic conductor: \(2.08050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (0:\ ),\ 0.225 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.510359475 - 1.201262229i\)
\(L(\frac12)\) \(\approx\) \(1.510359475 - 1.201262229i\)
\(L(1)\) \(\approx\) \(1.380697740 - 0.5599966527i\)
\(L(1)\) \(\approx\) \(1.380697740 - 0.5599966527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.793 - 0.608i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
11 \( 1 + (-0.130 + 0.991i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.991 - 0.130i)T \)
23 \( 1 + (-0.258 + 0.965i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.130 + 0.991i)T \)
59 \( 1 + (-0.991 - 0.130i)T \)
61 \( 1 + (0.130 + 0.991i)T \)
67 \( 1 + (0.793 - 0.608i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.965 + 0.258i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.965 + 0.258i)T \)
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

−24.46900975891323862106109538561, −23.25440836974859482321764677160, −22.11463615205571988281390200192, −21.67489904112820871101852194014, −20.93321176431877313720403534918, −19.92638183517281761404322310523, −19.00887598943246691835888334421, −18.39742958333217357505896252466, −17.255889391719659095771636005467, −16.17256954603462684049685251290, −15.56320619705861290283848634503, −14.31432994197211622677265151259, −14.0095841944184187909618658162, −13.158524706711244121908180633685, −11.61299124795955607420826984642, −10.712396100082463760510838133854, −10.02314566824788321905333147649, −8.967281726471130099110107102161, −8.28873645765212474505558452333, −6.96186274764516403343347397622, −6.0848246727857483786247061998, −4.82416761682230376179108720745, −3.65834710714087181430678213273, −2.81269230314329725631240142165, −1.75845080592112298923857935968, 1.072192145884104377919072427399, 2.09244012752559810196964911677, 3.16178077932135305730072288292, 4.520836918741525806078499767101, 5.54992149558825782941023632389, 6.73164683037328995637943126406, 7.6913541585425913643949017777, 8.58622780500621904028058021787, 9.45817905849096541037851104472, 10.18883791895433108383366069549, 11.78751131094782898680950414171, 12.54061243473071612817404351322, 13.49618237983311940718973214944, 13.819786611663427739814800548286, 15.26029424133752898547870752992, 15.73091383754325915592712089497, 17.20831196105676486061217961265, 17.846274823287311453167862407402, 18.513011530863404149738980770891, 20.003940584704696714594103550474, 20.1453803465031956210383884925, 20.98252274680691666351323170568, 22.07235694284373436879041903910, 23.1137298178965437666878899235, 24.03696352591474957342993791512

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