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Properties

Label 2-930-155.37-c1-0-13
Degree $2$
Conductor $930$
Sign $0.0127 - 0.999i$
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 + (0.795 + 2.08i)5-s + (0.866 − 0.500i)6-s + (−0.0433 + 0.161i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.915 + 2.04i)10-s + (3.10 + 1.79i)11-s + (0.965 + 0.258i)12-s + (−5.07 + 1.36i)13-s + (−0.145 + 0.0838i)14-s + (2.22 − 0.227i)15-s − 1.00·16-s + (7.84 + 2.10i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.149 − 0.557i)3-s + 0.500i·4-s + (0.355 + 0.934i)5-s + (0.353 − 0.204i)6-s + (−0.0164 + 0.0612i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.289 + 0.645i)10-s + (0.935 + 0.540i)11-s + (0.278 + 0.0747i)12-s + (−1.40 + 0.377i)13-s + (−0.0388 + 0.0224i)14-s + (0.574 − 0.0586i)15-s − 0.250·16-s + (1.90 + 0.509i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0127 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0127 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0127 - 0.999i$
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: \((2,\ 930,\ (\ :1/2),\ 0.0127 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55133 + 1.53161i\)
\(L(\frac12)\) \(\approx\) \(1.55133 + 1.53161i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.795 - 2.08i)T \)
31 \( 1 + (-4.42 - 3.37i)T \)
good7 \( 1 + (0.0433 - 0.161i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.10 - 1.79i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.07 - 1.36i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-7.84 - 2.10i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.88 - 1.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.44 + 1.44i)T + 23iT^{2} \)
29 \( 1 + 4.55T + 29T^{2} \)
37 \( 1 + (-8.44 - 2.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.57 - 7.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.935 + 3.48i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.62 + 6.62i)T + 47iT^{2} \)
53 \( 1 + (1.03 - 0.277i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-9.81 + 5.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.57iT - 61T^{2} \)
67 \( 1 + (1.98 - 0.532i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.481 - 0.833i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.70 + 0.457i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.34 - 7.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.5 + 3.90i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + (6.21 + 6.21i)T + 97iT^{2} \)
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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

−9.975266449535184789289693694358, −9.668801929902353895526504682450, −8.282798212418670246986194136334, −7.53673496568820273374246837902, −6.80999267681283405870124911495, −6.17915324261186092205009072306, −5.18207893686331439644402504485, −3.95078215158139480497120082124, −2.92496211139374980196735369941, −1.80692432892557075461551688455, 0.892841145389919064516143591281, 2.39913505202638760956014167137, 3.56710727478088688204427940779, 4.48492680519907349736556064580, 5.35324446997955073356237624004, 5.98001318809428095173886829872, 7.41212821446618269797049889092, 8.329783615616399586522561355456, 9.520383138080278857906546894713, 9.621543024320512182539483247119

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