Properties

Label 1-195-195.107-r0-0-0
Degree $1$
Conductor $195$
Sign $0.680 + 0.732i$
Analytic cond. $0.905576$
Root an. cond. $0.905576$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)11-s + 14-s + (−0.5 + 0.866i)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 + (0.866 + 0.5i)28-s + (−0.5 + 0.866i)29-s + 31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 − 0.866i)11-s + 14-s + (−0.5 + 0.866i)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 + (0.866 + 0.5i)28-s + (−0.5 + 0.866i)29-s + 31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.680 + 0.732i$
Analytic conductor: \(0.905576\)
Root analytic conductor: \(0.905576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 195,\ (0:\ ),\ 0.680 + 0.732i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.847427166 + 0.8049111981i\)
\(L(\frac12)\) \(\approx\) \(1.847427166 + 0.8049111981i\)
\(L(1)\) \(\approx\) \(1.670074004 + 0.5160807222i\)
\(L(1)\) \(\approx\) \(1.670074004 + 0.5160807222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 - T \)
83 \( 1 + iT \)
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

−27.12554415882072718791891122108, −25.72014932138863024000783093863, −24.59371561310857880219519690570, −24.14648203315985717255677477693, −22.85941859088928494685801415387, −22.1693618323211885411557815902, −21.21098714000030631804377396766, −20.33472931955517856179964586465, −19.538244969472286387272311312038, −18.27187678865469108060246801456, −17.418614583222382344795505853, −15.76226558215133051234970170067, −15.117321937197969248085552145335, −14.09817513510775893405468481531, −13.19948800864554583591205074001, −11.86689967136477097001004895843, −11.49433536066656426132819871732, −10.1119468887298005174690241369, −9.053852699020082618059155823209, −7.49319682478844020339172153974, −6.30437035958419461955433426142, −5.03472822446031094322483228151, −4.2563337053881008551967286987, −2.67123703967928618649418847369, −1.5954171398544974959560426946, 1.80899619790097582398272143542, 3.47149666979144673921126999425, 4.4248880421934798916453259452, 5.628829272969401842679235942231, 6.684270259603811435158903090587, 7.87658880276802879718778721684, 8.72115524261237620254068545936, 10.56379333771922540089726449899, 11.481003012937317871055346069100, 12.46248682501276238911640774111, 13.78585734500309379709392973757, 14.23930517980243750703974894920, 15.35967675764456680791427164176, 16.44152014421072811567305782721, 17.22010587727101183467281496359, 18.26841582832154352534866389136, 19.77968957342914750890585093843, 20.65076079803296229340895062478, 21.58483856269932529515891721572, 22.400940167498415768318675301391, 23.44435710968433638397571079448, 24.37025449877509557611263065541, 24.78172179217791392301870184849, 26.255447419213804526112000891942, 26.789314977221709394437401407264

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