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Properties

Label 1-14e2-196.103-r0-0-0
Degree $1$
Conductor $196$
Sign $0.453 + 0.891i$
Analytic cond. $0.910220$
Root an. cond. $0.910220$
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  + (0.365 + 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.365 + 0.930i)33-s + (0.826 + 0.563i)37-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)3-s + (0.988 + 0.149i)5-s + (−0.733 + 0.680i)9-s + (0.733 + 0.680i)11-s + (0.222 − 0.974i)13-s + (0.222 + 0.974i)15-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (−0.900 + 0.433i)29-s + (−0.5 + 0.866i)31-s + (−0.365 + 0.930i)33-s + (0.826 + 0.563i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.453 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283286330 + 0.7873422983i\)
\(L(\frac12)\) \(\approx\) \(1.283286330 + 0.7873422983i\)
\(L(1)\) \(\approx\) \(1.255043262 + 0.4553206887i\)
\(L(1)\) \(\approx\) \(1.255043262 + 0.4553206887i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.365 + 0.930i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.733 - 0.680i)T \)
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

−26.58908156382296470034717516337, −25.6569720988163374216323530397, −24.91970486459553111270381330628, −24.22037263667219383168673662536, −23.17898021630597669787058655259, −22.001115438110381385835962885348, −21.04935856554787919619805659447, −20.1221118980440599056891645217, −18.95378942477095782460146782300, −18.35958865570711738094051321613, −17.19588294732533609677519150373, −16.49520694270274627291486329161, −14.79112434024449803337263491729, −13.88504240862085164192958779610, −13.3727034811627496881634844043, −12.11641160363198574024492239283, −11.18823158693894911018463292389, −9.510445026152584152524606484270, −8.92408308850962049445107519108, −7.56889014357312765144757580905, −6.428731610120600870798632646465, −5.65402337089482940319895398411, −3.83545034719545207623949920725, −2.35189695033081414657809997330, −1.32867384581257188897939066444, 1.89812381673755897376992765892, 3.16400976734237504100032162868, 4.463969131902142705518550159505, 5.5787333075841302199135784448, 6.7385725199190002277274388275, 8.37549204886477823556418364848, 9.26261112082293476589576626572, 10.24959705771224047246445954972, 10.93346739386452186074780620092, 12.58043487930190001547887148197, 13.57486102773147672823697996870, 14.74009478085074920710540662773, 15.20171472273467497933708566616, 16.68492820931990266798753493247, 17.31904448163467266841279841568, 18.40981330476420198513829688337, 19.89072016891680475658263013247, 20.411118152416788273665102767583, 21.651984442707167508825016825087, 22.074496883344102155522627148933, 23.12273353840799426991366781443, 24.61093139214310397315700899666, 25.52118962786286117631427786363, 26.00445118121614823130514426037, 27.15058349194105773739094497027

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