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Properties

Label 2-930-93.26-c1-0-8
Degree $2$
Conductor $930$
Sign $0.733 - 0.679i$
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 + (1.41 − 0.993i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.993 − 1.41i)6-s + (−1.64 + 2.85i)7-s + i·8-s + (1.02 − 2.81i)9-s + (0.5 + 0.866i)10-s + (2.34 + 4.05i)11-s + (−1.41 + 0.993i)12-s + (−3.57 + 2.06i)13-s + (2.85 + 1.64i)14-s + (−0.731 + 1.56i)15-s + 16-s + (−3.02 + 5.23i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.818 − 0.573i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.405 − 0.579i)6-s + (−0.622 + 1.07i)7-s + 0.353i·8-s + (0.341 − 0.939i)9-s + (0.158 + 0.273i)10-s + (0.706 + 1.22i)11-s + (−0.409 + 0.286i)12-s + (−0.991 + 0.572i)13-s + (0.762 + 0.440i)14-s + (−0.188 + 0.405i)15-s + 0.250·16-s + (−0.732 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15590 + 0.453342i\)
\(L(\frac12)\) \(\approx\) \(1.15590 + 0.453342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 0.993i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (5.16 - 2.07i)T \)
good7 \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.34 - 4.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.57 - 2.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.02 - 5.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.11 - 1.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.244T + 23T^{2} \)
29 \( 1 - 2.05T + 29T^{2} \)
37 \( 1 + (7.39 + 4.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.83 + 3.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 5.79i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.99iT - 47T^{2} \)
53 \( 1 + (-0.996 - 1.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.641 + 0.370i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.30iT - 61T^{2} \)
67 \( 1 + (-1.90 - 3.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.36 + 4.83i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.290 + 0.167i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.1 + 6.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.42 + 2.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.149T + 89T^{2} \)
97 \( 1 + 7.67T + 97T^{2} \)
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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

−9.984213935728419108733806860706, −9.205992588407981026130285770451, −8.834187873565373766227145735080, −7.68465043526848920697856655778, −6.88487148707152746715496964055, −5.99877019432105348417103686302, −4.47843511250559717679503668204, −3.68367459529418699164214373171, −2.47064654485586960261004804682, −1.83828556198080323185867750760, 0.51523926078866973352794764023, 2.81030243919238235241448331362, 3.76468143948133809965460353517, 4.51984443222953617020954842891, 5.52319012366014996519416060606, 6.90629642761812434592701106602, 7.33960313234504393020568666523, 8.349921529864388192926344293384, 9.062832602691601844838837961192, 9.715189595289375442201385091924

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