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Properties

Label 2-930-155.149-c1-0-1
Degree $2$
Conductor $930$
Sign $0.151 - 0.988i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  − i·2-s + (0.866 − 0.5i)3-s − 4-s + (−0.196 − 2.22i)5-s + (−0.5 − 0.866i)6-s + (−3.49 + 2.01i)7-s + i·8-s + (0.499 − 0.866i)9-s + (−2.22 + 0.196i)10-s + (−2.26 + 3.91i)11-s + (−0.866 + 0.5i)12-s + (2.75 + 1.58i)13-s + (2.01 + 3.49i)14-s + (−1.28 − 1.83i)15-s + 16-s + (−6.46 + 3.73i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (−0.0877 − 0.996i)5-s + (−0.204 − 0.353i)6-s + (−1.32 + 0.763i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.704 + 0.0620i)10-s + (−0.682 + 1.18i)11-s + (−0.249 + 0.144i)12-s + (0.762 + 0.440i)13-s + (0.539 + 0.934i)14-s + (−0.331 − 0.472i)15-s + 0.250·16-s + (−1.56 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.151 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293372 + 0.251719i\)
\(L(\frac12)\) \(\approx\) \(0.293372 + 0.251719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.196 + 2.22i)T \)
31 \( 1 + (-5.11 + 2.19i)T \)
good7 \( 1 + (3.49 - 2.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.26 - 3.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.75 - 1.58i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.46 - 3.73i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.03 + 3.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.80iT - 23T^{2} \)
29 \( 1 + 1.85T + 29T^{2} \)
37 \( 1 + (4.95 - 2.86i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.959 - 1.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-9.43 + 5.44i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.15iT - 47T^{2} \)
53 \( 1 + (10.6 + 6.13i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.14 - 5.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.50T + 61T^{2} \)
67 \( 1 + (-0.256 - 0.148i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.748 - 1.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.36 + 4.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.40 - 7.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.7 + 7.92i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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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

−10.08520434262866724603905749128, −9.213425432169063716037750163638, −8.952544404647298881762415992131, −8.026294622835799670071724949287, −6.81405440331325559165906917689, −5.94397248738219192620838678795, −4.71070929750444183489171547447, −3.90703502895406170473559618155, −2.64552727198338687806848374065, −1.76038041691324200601549446880, 0.16071372375392966840064132537, 2.75666829466712920609357556322, 3.45287818887052857233203168880, 4.37416298518329284302358887506, 5.89043158782363261937485242360, 6.48091323290019001186986423662, 7.24457404152297123212180446897, 8.169075121577995284565227860013, 8.933384829731264818636523883298, 9.872740606132179309546796981893

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