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Properties

Label 2-931-133.4-c1-0-40
Degree $2$
Conductor $931$
Sign $0.930 + 0.365i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.152 + 0.866i)2-s + (0.5 + 0.181i)3-s + (1.15 − 0.419i)4-s + (−2.37 − 0.866i)5-s + (−0.0812 + 0.460i)6-s + (1.41 + 2.45i)8-s + (−2.08 − 1.74i)9-s + (0.386 − 2.19i)10-s + 3.41·11-s + 0.652·12-s + (0.918 − 5.21i)13-s + (−1.03 − 0.866i)15-s + (−0.0320 + 0.0269i)16-s + (−1.26 + 1.06i)17-s + (1.19 − 2.06i)18-s + (1.81 − 3.96i)19-s + ⋯
L(s)  = 1  + (0.107 + 0.612i)2-s + (0.288 + 0.105i)3-s + (0.576 − 0.209i)4-s + (−1.06 − 0.387i)5-s + (−0.0331 + 0.188i)6-s + (0.501 + 0.868i)8-s + (−0.693 − 0.582i)9-s + (0.122 − 0.693i)10-s + 1.02·11-s + 0.188·12-s + (0.254 − 1.44i)13-s + (−0.266 − 0.223i)15-s + (−0.00802 + 0.00673i)16-s + (−0.307 + 0.257i)17-s + (0.281 − 0.487i)18-s + (0.417 − 0.908i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75896 - 0.332731i\)
\(L(\frac12)\) \(\approx\) \(1.75896 - 0.332731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (-1.81 + 3.96i)T \)
good2 \( 1 + (-0.152 - 0.866i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-0.5 - 0.181i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (2.37 + 0.866i)T + (3.83 + 3.21i)T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + (-0.918 + 5.21i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.26 - 1.06i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.305 + 1.73i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.25 + 1.18i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.971 - 1.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.418 - 0.725i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.779 - 4.42i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-3.67 + 3.08i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.549 + 0.460i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-5.73 + 2.08i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-8.24 + 6.91i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.762 - 4.32i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.46 + 13.9i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.5 - 8.84i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (7.06 + 2.57i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (5.33 - 4.47i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.25 + 2.17i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.14 - 0.780i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.71 + 0.623i)T + (74.3 + 62.3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00390367874385281272760027984, −8.804059915992392206182648677111, −8.330646705840104709898897805503, −7.50666766195817871569629872129, −6.60499225647594035642029552094, −5.82649997471316235068369712251, −4.77952533499242726447531619697, −3.68706838977436827113400672847, −2.70710170294270356533955760872, −0.834233094562700456293448621358, 1.54707438891678737434215084951, 2.70397471427636802528390186429, 3.71606929513049393382139341487, 4.30342686584667059347174554605, 5.93339780571052604048495424308, 6.95619243952164067424084215506, 7.47130943841911868485832343677, 8.440073906465280110102899205151, 9.261524799591986999015674879130, 10.34337092364824456408666981609

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