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Properties

Label 2-930-155.37-c1-0-21
Degree $2$
Conductor $930$
Sign $0.991 + 0.129i$
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 + (−2.17 + 0.511i)5-s + (0.866 − 0.500i)6-s + (0.572 − 2.13i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.90 − 1.17i)10-s + (3.84 + 2.21i)11-s + (0.965 + 0.258i)12-s + (4.83 − 1.29i)13-s + (1.91 − 1.10i)14-s + (−0.0696 + 2.23i)15-s − 1.00·16-s + (−2.89 − 0.776i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.149 − 0.557i)3-s + 0.500i·4-s + (−0.973 + 0.228i)5-s + (0.353 − 0.204i)6-s + (0.216 − 0.807i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.601 − 0.372i)10-s + (1.15 + 0.668i)11-s + (0.278 + 0.0747i)12-s + (1.34 − 0.359i)13-s + (0.511 − 0.295i)14-s + (−0.0179 + 0.577i)15-s − 0.250·16-s + (−0.702 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.991 + 0.129i$
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: \((2,\ 930,\ (\ :1/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00491 - 0.129962i\)
\(L(\frac12)\) \(\approx\) \(2.00491 - 0.129962i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 0.511i)T \)
31 \( 1 + (-4.48 - 3.30i)T \)
good7 \( 1 + (-0.572 + 2.13i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.84 - 2.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.83 + 1.29i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (2.89 + 0.776i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.25 + 1.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.93 + 5.93i)T + 23iT^{2} \)
29 \( 1 - 5.40T + 29T^{2} \)
37 \( 1 + (-7.62 - 2.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.52 + 4.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.94 + 7.24i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.55 - 5.55i)T + 47iT^{2} \)
53 \( 1 + (-2.44 + 0.655i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-8.32 + 4.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 5.10iT - 61T^{2} \)
67 \( 1 + (12.0 - 3.22i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.91 - 13.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.88 - 2.64i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.194 - 0.337i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.7 - 3.95i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 8.22T + 89T^{2} \)
97 \( 1 + (0.961 + 0.961i)T + 97iT^{2} \)
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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

−10.16861630520393478745203862876, −8.761179648973047316304177328487, −8.292979087749257682208639750934, −7.29155749549793249557520828955, −6.79627523646569874221456213577, −5.98464915443222264886409008489, −4.32320649306947699838930960692, −4.10189096225692462595853909476, −2.77609470105690532422979870027, −0.992853547819617741499691032012, 1.29828689137526277583120059532, 2.89666706597350567621247663317, 3.94810653860012080656424421199, 4.32948268033460584860620410068, 5.73709378097324728403574762264, 6.29404963096608275907121278548, 7.76459996034909374809048707262, 8.695999489515572669771404163029, 9.092398635979166402030248325906, 10.17747449299228608476361517317

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