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Properties

Label 1-41-41.32-r0-0-0
Degree $1$
Conductor $41$
Sign $-0.331 - 0.943i$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
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  − 2-s − i·3-s + 4-s − 5-s + i·6-s − i·7-s − 8-s − 9-s + 10-s − i·11-s − i·12-s − i·13-s + i·14-s + i·15-s + 16-s + i·17-s + ⋯
L(s)  = 1  − 2-s − i·3-s + 4-s − 5-s + i·6-s − i·7-s − 8-s − 9-s + 10-s − i·11-s − i·12-s − i·13-s + i·14-s + i·15-s + 16-s + i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2486102712 - 0.3506698453i\)
\(L(\frac12)\) \(\approx\) \(0.2486102712 - 0.3506698453i\)
\(L(1)\) \(\approx\) \(0.4861789541 - 0.2843950774i\)
\(L(1)\) \(\approx\) \(0.4861789541 - 0.2843950774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - iT \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - iT \)
43 \( 1 - iT \)
47 \( 1 + iT \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 + T \)
71 \( 1 + iT \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + iT \)
89 \( 1 + T \)
97 \( 1 + iT \)
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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

−35.211023529428627598366547763731, −34.24566151902438354557631556129, −33.31412305165847698869368733050, −31.70327254931063325891423820511, −30.76191454041382427224635965968, −28.73941659877253690923167462210, −28.05486342286608498275344502147, −27.11496818325212077078713351297, −26.105672629754121168417760338322, −24.917666120948777502061201964413, −23.302864918034813652947631393269, −21.79260168284993643842555357907, −20.54919207249746418983893132127, −19.50161935764314440399574809097, −18.22598538001553404278044771995, −16.68305503874158915427224030530, −15.623943409721627092104947702646, −14.9113438094063477935931128003, −12.02034479521725431100289327613, −11.18822389224312707624498109052, −9.54012583492575939633268056846, −8.65534664980172020111850648793, −6.96850229428483280011237096520, −4.81654558690715959083970568338, −2.806323676719557206251645586629, 0.91311920800109113475076478007, 3.27457351164645066725475529448, 6.25090562681315849791564434387, 7.68724011132996447771734578901, 8.32712002700972604304172991021, 10.53766023516713971759692744391, 11.617156277717613156507671898500, 13.03449532024492811858888460748, 14.840988277023686629263322850969, 16.44537581568818779034963577079, 17.44793261061035752323324695443, 18.906704181092478251955543909504, 19.555789843082231859022112494096, 20.66900227868369895524187131962, 23.00551090983796676121213293022, 23.98112218177773552515617277434, 25.01987119461017154344171725225, 26.46990395092926721578562961735, 27.308578274739262193911812546210, 28.7087406212539138064075451996, 29.85765705746507075865166003036, 30.55863650277040576455174790734, 32.2240063731091488246078047306, 33.915815033107170339035341365457, 35.0613138563150031998615296661

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