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Properties

Label 1-41-41.38-r1-0-0
Degree $1$
Conductor $41$
Sign $0.916 + 0.400i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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 + (0.707 + 0.707i)3-s − 4-s + i·5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + i·8-s + i·9-s + 10-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s + (0.707 − 0.707i)17-s + ⋯
L(s)  = 1  − i·2-s + (0.707 + 0.707i)3-s − 4-s + i·5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + i·8-s + i·9-s + 10-s + (−0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)14-s + (−0.707 + 0.707i)15-s + 16-s + (0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.916 + 0.400i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.654415799 + 0.3461397334i\)
\(L(\frac12)\) \(\approx\) \(1.654415799 + 0.3461397334i\)
\(L(1)\) \(\approx\) \(1.281838162 + 0.02537637171i\)
\(L(1)\) \(\approx\) \(1.281838162 + 0.02537637171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 - T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−34.71154877701559415008358491101, −33.25574960652598169657735675917, −32.31962197299286542667979421216, −31.213871064385457884041403073187, −30.258294169210918426985976209547, −28.45675908264593196478781158872, −27.20739517731523760148956295571, −25.885752801809969279145520724860, −24.99894896416926185739113380729, −23.89489124687093414781970740016, −23.25475785440931785554564947331, −21.00015394812015760808513692700, −20.02712463113146832022211585326, −18.31294829359159734231456607907, −17.41519464380640435787089862163, −15.9713772447094487864215654927, −14.58889756294935751720199545312, −13.447887148588225182340795652207, −12.47298992097546472112198489634, −9.8532992293281800796540701319, −8.196038556987242080069239164098, −7.7053018548101531309235125708, −5.73711177473185580202049832437, −4.06768141736943721923885370821, −1.155496280544986017859835479263, 2.327949240737646937328996044873, 3.52062260354254635867434182418, 5.28057546808117211819245304927, 8.016752678989336757533032521635, 9.29914519036515813645675325711, 10.66306047777142418809093468779, 11.59871593663500546830798393458, 13.70426141505741516930376191993, 14.47652242227329456190610619252, 15.953704647042803877678699865370, 18.17868173808686175701286093763, 18.85531262404281140166084395749, 20.3256979653290873741024944295, 21.47769196805310118278175887, 22.03959038368858674925919154355, 23.6784601905785630169942840112, 25.6362647291497004219927928268, 26.655789943257901780129124978390, 27.531225254384246357069473533336, 28.78189929460615245459156466833, 30.30633742747723513310344973366, 31.0712183129297192777129629630, 31.96103957061084051003806840962, 33.44697103932348746313336592652, 34.64621371218631705327054168470

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