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Properties

Label 2-930-155.37-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.260 + 0.965i$
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 + (−0.258 + 0.965i)3-s + 1.00i·4-s + (−2.08 + 0.808i)5-s + (−0.866 + 0.500i)6-s + (−0.0488 + 0.182i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.04 − 0.902i)10-s + (−5.38 − 3.11i)11-s + (−0.965 − 0.258i)12-s + (1.35 − 0.364i)13-s + (−0.163 + 0.0943i)14-s + (−0.241 − 2.22i)15-s − 1.00·16-s + (6.65 + 1.78i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (−0.932 + 0.361i)5-s + (−0.353 + 0.204i)6-s + (−0.0184 + 0.0688i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.646 − 0.285i)10-s + (−1.62 − 0.938i)11-s + (−0.278 − 0.0747i)12-s + (0.377 − 0.101i)13-s + (−0.0436 + 0.0252i)14-s + (−0.0622 − 0.573i)15-s − 0.250·16-s + (1.61 + 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.260 + 0.965i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0227129 - 0.0296519i\)
\(L(\frac12)\) \(\approx\) \(0.0227129 - 0.0296519i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (2.08 - 0.808i)T \)
31 \( 1 + (-4.39 + 3.42i)T \)
good7 \( 1 + (0.0488 - 0.182i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.38 + 3.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.35 + 0.364i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-6.65 - 1.78i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.32 - 3.07i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.32 + 6.32i)T + 23iT^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
37 \( 1 + (4.10 + 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.781 + 1.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.471 - 1.76i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.40 + 1.40i)T + 47iT^{2} \)
53 \( 1 + (0.133 - 0.0359i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.717 + 0.414i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + (3.81 - 1.02i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.11 + 5.38i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (14.8 - 3.97i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.50 + 2.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.9 - 4.01i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 5.85T + 89T^{2} \)
97 \( 1 + (11.8 + 11.8i)T + 97iT^{2} \)
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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

−10.52974280306462309716225981796, −10.20138498674725005671769566196, −8.531043734828602365566747480352, −8.169328782092263599910770124643, −7.41709394238738433847092304654, −6.02133787175064591097981941843, −5.65920447069023295553613034859, −4.36275177040227133510948801659, −3.65042063315144277481200975589, −2.68138741611222544491797423054, 0.01450423061757147497449903452, 1.68891843391921369351089095413, 2.94743227634538233577360292636, 4.01804712209530360348498407335, 5.04874440416522124535658453629, 5.71897385670208468102256750134, 7.10124385722459003872458819301, 7.69906918211068760329844605271, 8.453044074319850689082519282555, 9.703135242383626099452712469499

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