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Properties

Label 1-1205-1205.1053-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.913 + 0.406i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + i·6-s + (0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 − 0.991i)11-s + (−0.965 − 0.258i)12-s + (0.991 + 0.130i)13-s + (−0.130 + 0.991i)14-s + (0.5 + 0.866i)16-s + (0.382 + 0.923i)17-s + (0.258 + 0.965i)18-s + (0.991 − 0.130i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.5i)4-s + i·6-s + (0.991 − 0.130i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.5i)9-s + (−0.130 − 0.991i)11-s + (−0.965 − 0.258i)12-s + (0.991 + 0.130i)13-s + (−0.130 + 0.991i)14-s + (0.5 + 0.866i)16-s + (0.382 + 0.923i)17-s + (0.258 + 0.965i)18-s + (0.991 − 0.130i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.913 + 0.406i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.913 + 0.406i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.150847914 + 0.4567256798i\)
\(L(\frac12)\) \(\approx\) \(2.150847914 + 0.4567256798i\)
\(L(1)\) \(\approx\) \(1.404522930 + 0.3417721506i\)
\(L(1)\) \(\approx\) \(1.404522930 + 0.3417721506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
13 \( 1 + (0.991 + 0.130i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
19 \( 1 + (0.991 - 0.130i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.793 - 0.608i)T \)
37 \( 1 + (-0.991 + 0.130i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
71 \( 1 + (0.793 + 0.608i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.793 + 0.608i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.80775666133284366084119689312, −20.373846808054274813385734685421, −20.000207791905658279535223158692, −18.658887475701075112465207638664, −18.35907557483198297938779941949, −17.68332597064885769212072994698, −16.48581698793591667107741278245, −15.68837644741622734085783083293, −14.66661896771387973928903838043, −14.059724711598800889486433219934, −13.44908737617964377292560142539, −12.375161290250867111662555553389, −11.82687933347506595040616464651, −10.61918954970887069686964756983, −10.28647040797607621901562210985, −9.10591534072371435458954548222, −8.743029728503510957004765220995, −7.76262494548129265324708556693, −7.17715678452170801209728454218, −5.295796193156334447969684877281, −4.69179664000494987738754347686, −3.73236663862673915016991487785, −2.9349570706759362875128018770, −1.94226923841139942381254708904, −1.271433981911211568308326164915, 1.056732495715636217786488139110, 1.798907726492707760989418492772, 3.41909110509918537855007982210, 3.9939950014216828206820926154, 5.22284774745577716492594939667, 6.00275175767703315034113659659, 6.98379628545554483567587057163, 8.03026503043379379750626944552, 8.16583703192220541499141967197, 9.05274863643542542135926264307, 9.925032010315926020724567609491, 10.87474079306119766402921272816, 11.85609430291393781182107017650, 13.17526677436925017556171997203, 13.74314506933284868377786537285, 14.16808747683136585705568956947, 15.16940517296619414019644095632, 15.60362345561345252849562697533, 16.58340671102544548949072887723, 17.37702854463118010710814809531, 18.41005970593451669473181366360, 18.5585856123576987450953612648, 19.59774736514253949021895364304, 20.3187820071206733573783467204, 21.36202653326787755976149638510

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