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Properties

Label 2-930-15.2-c1-0-11
Degree $2$
Conductor $930$
Sign $0.901 - 0.432i$
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  + (−0.707 + 0.707i)2-s + (−0.404 − 1.68i)3-s − 1.00i·4-s + (−2.01 + 0.959i)5-s + (1.47 + 0.904i)6-s + (−2.95 − 2.95i)7-s + (0.707 + 0.707i)8-s + (−2.67 + 1.36i)9-s + (0.749 − 2.10i)10-s + 4.99i·11-s + (−1.68 + 0.404i)12-s + (−0.576 + 0.576i)13-s + 4.18·14-s + (2.43 + 3.01i)15-s − 1.00·16-s + (1.92 − 1.92i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.233 − 0.972i)3-s − 0.500i·4-s + (−0.903 + 0.429i)5-s + (0.602 + 0.369i)6-s + (−1.11 − 1.11i)7-s + (0.250 + 0.250i)8-s + (−0.890 + 0.454i)9-s + (0.236 − 0.666i)10-s + 1.50i·11-s + (−0.486 + 0.116i)12-s + (−0.159 + 0.159i)13-s + 1.11·14-s + (0.628 + 0.777i)15-s − 0.250·16-s + (0.467 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.901 - 0.432i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576086 + 0.131171i\)
\(L(\frac12)\) \(\approx\) \(0.576086 + 0.131171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.404 + 1.68i)T \)
5 \( 1 + (2.01 - 0.959i)T \)
31 \( 1 + T \)
good7 \( 1 + (2.95 + 2.95i)T + 7iT^{2} \)
11 \( 1 - 4.99iT - 11T^{2} \)
13 \( 1 + (0.576 - 0.576i)T - 13iT^{2} \)
17 \( 1 + (-1.92 + 1.92i)T - 17iT^{2} \)
19 \( 1 - 2.28iT - 19T^{2} \)
23 \( 1 + (2.35 + 2.35i)T + 23iT^{2} \)
29 \( 1 - 7.87T + 29T^{2} \)
37 \( 1 + (-5.67 - 5.67i)T + 37iT^{2} \)
41 \( 1 + 6.89iT - 41T^{2} \)
43 \( 1 + (-7.51 + 7.51i)T - 43iT^{2} \)
47 \( 1 + (4.39 - 4.39i)T - 47iT^{2} \)
53 \( 1 + (3.78 + 3.78i)T + 53iT^{2} \)
59 \( 1 + 8.23T + 59T^{2} \)
61 \( 1 - 2.57T + 61T^{2} \)
67 \( 1 + (-8.10 - 8.10i)T + 67iT^{2} \)
71 \( 1 - 0.808iT - 71T^{2} \)
73 \( 1 + (-6.98 + 6.98i)T - 73iT^{2} \)
79 \( 1 - 6.04iT - 79T^{2} \)
83 \( 1 + (-5.46 - 5.46i)T + 83iT^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + (-9.00 - 9.00i)T + 97iT^{2} \)
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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

−10.19172242536041663633157857219, −9.305941998870191155335726696075, −7.974491068430308731099220157442, −7.57834008931859659250778684413, −6.75702600703401116345442743979, −6.44899543684616056859899260571, −4.91237454300967582830147221395, −3.82631093575941154356780899032, −2.49159238477851260197852828022, −0.803080448370472002326957291679, 0.51572919771924323715021124249, 2.91897485748206159417985511424, 3.37830364177227687648009027148, 4.50888038005017530688994561043, 5.68127270334144290040618870612, 6.35483431149865293028098000015, 7.937731198930408993426331750345, 8.565244676800635734154239346715, 9.257890044801759989089178271632, 9.853866847079836834653952687493

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