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Properties

Label 1-315-315.52-r0-0-0
Degree $1$
Conductor $315$
Sign $-0.874 - 0.484i$
Analytic cond. $1.46285$
Root an. cond. $1.46285$
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  + i·2-s − 4-s − i·8-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + 16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s − 31-s + i·32-s + (−0.5 − 0.866i)34-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + 16-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (0.5 + 0.866i)29-s − 31-s + i·32-s + (−0.5 − 0.866i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(1.46285\)
Root analytic conductor: \(1.46285\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 315,\ (0:\ ),\ -0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09986406173 + 0.3864275441i\)
\(L(\frac12)\) \(\approx\) \(-0.09986406173 + 0.3864275441i\)
\(L(1)\) \(\approx\) \(0.5476191217 + 0.4131705338i\)
\(L(1)\) \(\approx\) \(0.5476191217 + 0.4131705338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + iT \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + iT \)
71 \( 1 + T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 - T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−24.36161062347088193738861057201, −23.813047240083392199461311394085, −22.59955837897227143050478870010, −21.866934522819973949122566118519, −21.18734691858177202165791162681, −20.11952542354333507097387027716, −19.415246998600334156513029288815, −18.54439395418444678167812585757, −17.67646489660034211474575990290, −16.7122032196494047213327676560, −15.535114156256894872511887724603, −14.30671912709107262779205689474, −13.54964576056953740920791888066, −12.64079399338327900310449118683, −11.604298480062309607139679664690, −10.883853159927178460001695309099, −9.83486580237418036404535439460, −8.913742421189676397281569409224, −7.96179395737090452066707338755, −6.51902468529442313425066606341, −5.14121244509251203863517592585, −4.27014845363580054848526306526, −2.937170241364571691634877290528, −2.035472346811261024849521159769, −0.24224175551764560704649637256, 1.9793903130622046051412994511, 3.684290835710299225897725584238, 4.78483176922179666157125153738, 5.69508045061358709594413180272, 6.87623922865134742023160908080, 7.698940578001008347678167195360, 8.65570256455189995313111921952, 9.79181760950673002992256496486, 10.57589923226546728287388849950, 12.30506250412313975515962975364, 12.88920842501558235465166082712, 14.077834701931371231058627956305, 14.932874974773104076942776849756, 15.636164405984651883068745063907, 16.64390338579157739801564435804, 17.604312927976790779062349054375, 18.14044840660182999135500467069, 19.34719077648162528498524433210, 20.24482360894143363578732025100, 21.55604574807030241321552159793, 22.3174627378833035424496440334, 23.216864745014229833471842920270, 23.96732298264029909562657513422, 24.87289978373919511728313220564, 25.65525460935616539107897797392

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