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Properties

Label 2-30-5.2-c2-0-1
Degree $2$
Conductor $30$
Sign $0.938 - 0.344i$
Analytic cond. $0.817440$
Root an. cond. $0.904124$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (−4.89 − i)5-s + 2.44·6-s + (0.898 + 0.898i)7-s + (−2 + 2i)8-s − 2.99i·9-s + (−3.89 − 5.89i)10-s − 13.7·11-s + (2.44 + 2.44i)12-s + (12.7 − 12.7i)13-s + 1.79i·14-s + (−7.22 + 4.77i)15-s − 4·16-s + (15.8 + 15.8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (−0.979 − 0.200i)5-s + 0.408·6-s + (0.128 + 0.128i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (−0.389 − 0.589i)10-s − 1.25·11-s + (0.204 + 0.204i)12-s + (0.984 − 0.984i)13-s + 0.128i·14-s + (−0.481 + 0.318i)15-s − 0.250·16-s + (0.935 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(0.817440\)
Root analytic conductor: \(0.904124\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :1),\ 0.938 - 0.344i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.18438 + 0.210763i\)
\(L(\frac12)\) \(\approx\) \(1.18438 + 0.210763i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 + (4.89 + i)T \)
good7 \( 1 + (-0.898 - 0.898i)T + 49iT^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + (-12.7 + 12.7i)T - 169iT^{2} \)
17 \( 1 + (-15.8 - 15.8i)T + 289iT^{2} \)
19 \( 1 - 25.7iT - 361T^{2} \)
23 \( 1 + (-10.6 + 10.6i)T - 529iT^{2} \)
29 \( 1 + 25.7iT - 841T^{2} \)
31 \( 1 + 39.5T + 961T^{2} \)
37 \( 1 + (27 + 27i)T + 1.36e3iT^{2} \)
41 \( 1 - 17.7T + 1.68e3T^{2} \)
43 \( 1 + (12.4 - 12.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.30 - 9.30i)T + 2.20e3iT^{2} \)
53 \( 1 + (19.0 - 19.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 20iT - 3.48e3T^{2} \)
61 \( 1 - 15.1T + 3.72e3T^{2} \)
67 \( 1 + (48.0 + 48.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 6.20T + 5.04e3T^{2} \)
73 \( 1 + (37.2 - 37.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 115. iT - 6.24e3T^{2} \)
83 \( 1 + (-82.2 + 82.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 117. iT - 7.92e3T^{2} \)
97 \( 1 + (-81.9 - 81.9i)T + 9.40e3iT^{2} \)
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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

−16.54184593926340562574323050337, −15.52241242361220407705212307156, −14.62374178991327278899627010039, −13.09960359909425039160941375904, −12.34404933195726100747417669697, −10.67551265776307764616745642546, −8.349193146727985219638768480258, −7.69177153425512979347890941581, −5.66628190222639169842756353527, −3.56704809263669712972749869582, 3.26110301690589559140612416868, 4.91155903697854065693088417028, 7.33458185583414642348968177730, 8.978311315411257178099739521632, 10.67401117978856701201658190550, 11.57826483071142711382839765029, 13.12234872432572512204225987882, 14.26325281163067081574444291793, 15.50477606683840601489563549350, 16.24738200218909062813388601097

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