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Properties

Label 1-67-67.17-r0-0-0
Degree $1$
Conductor $67$
Sign $0.578 - 0.815i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
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  + (0.981 − 0.189i)2-s + (0.415 − 0.909i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.235 − 0.971i)6-s + (−0.327 + 0.945i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (−0.995 − 0.0950i)10-s + (0.235 + 0.971i)11-s + (0.0475 − 0.998i)12-s + (−0.888 + 0.458i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.723 − 0.690i)16-s + (0.928 + 0.371i)17-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (0.415 − 0.909i)3-s + (0.928 − 0.371i)4-s + (−0.959 − 0.281i)5-s + (0.235 − 0.971i)6-s + (−0.327 + 0.945i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (−0.995 − 0.0950i)10-s + (0.235 + 0.971i)11-s + (0.0475 − 0.998i)12-s + (−0.888 + 0.458i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.723 − 0.690i)16-s + (0.928 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347487822 - 0.6966008498i\)
\(L(\frac12)\) \(\approx\) \(1.347487822 - 0.6966008498i\)
\(L(1)\) \(\approx\) \(1.509166756 - 0.5407706813i\)
\(L(1)\) \(\approx\) \(1.509166756 - 0.5407706813i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.981 - 0.189i)T \)
3 \( 1 + (0.415 - 0.909i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (0.235 + 0.971i)T \)
13 \( 1 + (-0.888 + 0.458i)T \)
17 \( 1 + (0.928 + 0.371i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.888 - 0.458i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.786 + 0.618i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.995 + 0.0950i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.928 - 0.371i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (0.0475 - 0.998i)T \)
83 \( 1 + (0.723 - 0.690i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−32.08154326038879745905822262149, −31.471912412009818365543547358379, −30.20261130985255319585733481129, −29.278402893921571105949045810119, −27.41125495547129746157345044650, −26.74887775329777719406901071003, −25.59927554830038478692908559719, −24.29443789340600280580280934350, −23.04404157483063382839541077975, −22.39083725518079079472941892437, −21.10721430569917307632252638637, −20.08670066659871472979626927150, −19.250113451347588941451739195920, −16.65051603580664598211666210098, −16.26413468268140860473752570757, −14.78267474303920738943490727332, −14.23234787135375552633539010077, −12.653343679454295040625525159957, −11.21347914359954223415040605695, −10.27800649221087143206797423524, −8.241996117449735755780663083387, −7.09149316339999061101101116344, −5.25090274438995358114775427922, −3.86126304098878993580487454549, −3.14711062018433029913216489715, 1.999215649820783123250190944942, 3.412229731481298234886585295851, 5.05899313643259328673358307927, 6.69626705373173557454807722796, 7.733801708117496652402069524829, 9.40639129229899977446997802293, 11.61602908547877937154246343300, 12.25374717899445421623556581397, 13.14664521376112581881914758364, 14.78616731920620567750977515144, 15.31459531162537282458427165599, 17.0006125420997693727819410281, 18.89723899095715735820217089409, 19.521490618613911822136311803063, 20.56040364546044822609283203973, 21.99300916574044887903811548635, 23.18115547514517374117598560221, 23.9857471634186053915863622410, 24.97051584392329201955668811525, 25.89240967786543643930212673030, 27.857796279970376508434233828, 28.78296634995940561061800917384, 30.00083089376440532074284299445, 30.97507601518736305775682977617, 31.58951121711141859902472700268

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