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Properties

Label 2-30-15.2-c3-0-5
Degree $2$
Conductor $30$
Sign $0.0707 + 0.997i$
Analytic cond. $1.77005$
Root an. cond. $1.33043$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 1.41i)2-s + (−3.82 − 3.51i)3-s − 4.00i·4-s + (5.59 − 9.67i)5-s + (−10.3 + 0.430i)6-s + (8.77 + 8.77i)7-s + (−5.65 − 5.65i)8-s + (2.23 + 26.9i)9-s + (−5.77 − 21.6i)10-s + 47.3i·11-s + (−14.0 + 15.2i)12-s + (53.4 − 53.4i)13-s + 24.8·14-s + (−55.4 + 17.3i)15-s − 16.0·16-s + (−23.6 + 23.6i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.735 − 0.677i)3-s − 0.500i·4-s + (0.500 − 0.865i)5-s + (−0.706 + 0.0293i)6-s + (0.473 + 0.473i)7-s + (−0.250 − 0.250i)8-s + (0.0828 + 0.996i)9-s + (−0.182 − 0.683i)10-s + 1.29i·11-s + (−0.338 + 0.367i)12-s + (1.14 − 1.14i)13-s + 0.473·14-s + (−0.954 + 0.298i)15-s − 0.250·16-s + (−0.337 + 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0707 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0707 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.0707 + 0.997i$
Analytic conductor: \(1.77005\)
Root analytic conductor: \(1.33043\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :3/2),\ 0.0707 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.988840 - 0.921165i\)
\(L(\frac12)\) \(\approx\) \(0.988840 - 0.921165i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 1.41i)T \)
3 \( 1 + (3.82 + 3.51i)T \)
5 \( 1 + (-5.59 + 9.67i)T \)
good7 \( 1 + (-8.77 - 8.77i)T + 343iT^{2} \)
11 \( 1 - 47.3iT - 1.33e3T^{2} \)
13 \( 1 + (-53.4 + 53.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (23.6 - 23.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 48.0iT - 6.85e3T^{2} \)
23 \( 1 + (22.6 + 22.6i)T + 1.21e4iT^{2} \)
29 \( 1 + 2.24T + 2.43e4T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + (-141. - 141. i)T + 5.06e4iT^{2} \)
41 \( 1 - 417. iT - 6.89e4T^{2} \)
43 \( 1 + (280. - 280. i)T - 7.95e4iT^{2} \)
47 \( 1 + (31.8 - 31.8i)T - 1.03e5iT^{2} \)
53 \( 1 + (288. + 288. i)T + 1.48e5iT^{2} \)
59 \( 1 - 426.T + 2.05e5T^{2} \)
61 \( 1 - 343.T + 2.26e5T^{2} \)
67 \( 1 + (398. + 398. i)T + 3.00e5iT^{2} \)
71 \( 1 + 37.5iT - 3.57e5T^{2} \)
73 \( 1 + (180. - 180. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.01e3iT - 4.93e5T^{2} \)
83 \( 1 + (19.3 + 19.3i)T + 5.71e5iT^{2} \)
89 \( 1 - 417.T + 7.04e5T^{2} \)
97 \( 1 + (908. + 908. i)T + 9.12e5iT^{2} \)
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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

−16.32556088195958712171456167704, −14.88718022184649520576245502622, −13.21303616976678836706474292141, −12.67220544331805520935003062512, −11.47810276453807390722859229904, −10.06511303958754544558509500965, −8.198334252637749939076586811533, −6.09458833311637835187789825201, −4.88016124865950638104623870503, −1.64880579464028807044679322422, 3.82810423767403620997095334860, 5.68727237491536995151870098447, 6.85465769154588392568084651740, 9.006793057432993708393038306247, 10.82053569166419850710973187950, 11.45818027831815403784818062530, 13.54009737496438234577236926524, 14.34631383712694397340061794496, 15.71530318271582812914870856818, 16.62885548739644878853370135831

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