Properties

Label 1-931-931.113-r1-0-0
Degree $1$
Conductor $931$
Sign $-0.761 - 0.648i$
Analytic cond. $100.049$
Root an. cond. $100.049$
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  + (0.900 + 0.433i)2-s + (0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (0.222 + 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 18-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.222 + 0.974i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (0.222 + 0.974i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.900 − 0.433i)11-s + (−0.623 + 0.781i)12-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.222 + 0.974i)16-s + (0.623 − 0.781i)17-s − 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(100.049\)
Root analytic conductor: \(100.049\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 931,\ (1:\ ),\ -0.761 - 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4600643505 + 1.250143462i\)
\(L(\frac12)\) \(\approx\) \(-0.4600643505 + 1.250143462i\)
\(L(1)\) \(\approx\) \(1.242590830 + 0.8231566539i\)
\(L(1)\) \(\approx\) \(1.242590830 + 0.8231566539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.623 + 0.781i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−20.979229168676176026147417513178, −20.56563144434650630162448049603, −19.51560018889763804474095228588, −18.87094327926452664940758940901, −18.380036958192929923254481227233, −17.475898129746790952077028793929, −16.15162272998178180608009139894, −15.187603223306996778815847608460, −14.70156438443221027152587304419, −13.86661744754401379101417079239, −13.065418027280779605851736615072, −12.53837955734380292674659909995, −11.582727871757209398605142523803, −10.77296430308977840442782342295, −10.22367516196021588648085773622, −8.780382917038844377126206495305, −7.62604780823007805546277389078, −7.1166214061703962457908057213, −6.05509450423306482132314301921, −5.54382080981296993370650550864, −4.02647920661755002419027491338, −3.20217102266017422269079045567, −2.43324531155191765070814490546, −1.52942346040142361314079542233, −0.17555534761567233053184169195, 1.60561161956479082618725657686, 3.10832858448732538503345134142, 3.5975102182421181102010232295, 4.77247212861463262238445670397, 5.18951848371065853073788791022, 6.0185675667382590878909730985, 7.40086485491609195934009614465, 8.2462830067464999642405154199, 8.90504682143896255115999934219, 9.88850517178508701925665170204, 11.21540689186753462090697640317, 11.479490716501067720140069495784, 12.80827630409804399538829359211, 13.35896455639033937053435415022, 14.19612062884017348067140957772, 15.0570476936535685010403135996, 15.938447000632215670449922193749, 16.26073095343792370644906595699, 16.86750642434338580200186848651, 18.00040386847437377244322240884, 19.2271066942223997969145553337, 20.28175566342938247727871942248, 20.75407176129370538310050424154, 21.29824040047585097251806155813, 21.988784176001836980286068348766

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