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Properties

Label 2-87-29.25-c1-0-4
Degree $2$
Conductor $87$
Sign $-0.495 + 0.868i$
Analytic cond. $0.694698$
Root an. cond. $0.833485$
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  + (0.487 − 2.13i)2-s + (−0.900 + 0.433i)3-s + (−2.52 − 1.21i)4-s + (0.533 − 2.33i)5-s + (0.487 + 2.13i)6-s + (−1.21 + 0.586i)7-s + (−1.09 + 1.37i)8-s + (0.623 − 0.781i)9-s + (−4.73 − 2.27i)10-s + (2.18 + 2.74i)11-s + 2.80·12-s + (3.76 + 4.71i)13-s + (0.659 + 2.88i)14-s + (0.533 + 2.33i)15-s + (−1.08 − 1.36i)16-s + 3.05·17-s + ⋯
L(s)  = 1  + (0.344 − 1.51i)2-s + (−0.520 + 0.250i)3-s + (−1.26 − 0.608i)4-s + (0.238 − 1.04i)5-s + (0.199 + 0.872i)6-s + (−0.460 + 0.221i)7-s + (−0.388 + 0.487i)8-s + (0.207 − 0.260i)9-s + (−1.49 − 0.720i)10-s + (0.659 + 0.827i)11-s + 0.809·12-s + (1.04 + 1.30i)13-s + (0.176 + 0.772i)14-s + (0.137 + 0.603i)15-s + (−0.272 − 0.341i)16-s + 0.740·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $-0.495 + 0.868i$
Analytic conductor: \(0.694698\)
Root analytic conductor: \(0.833485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 87,\ (\ :1/2),\ -0.495 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498739 - 0.858895i\)
\(L(\frac12)\) \(\approx\) \(0.498739 - 0.858895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (5.14 + 1.58i)T \)
good2 \( 1 + (-0.487 + 2.13i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-0.533 + 2.33i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (1.21 - 0.586i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-2.18 - 2.74i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-3.76 - 4.71i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 + (5.07 + 2.44i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (0.225 + 0.989i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (1.80 - 7.90i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-3.42 + 4.29i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 6.85T + 41T^{2} \)
43 \( 1 + (-1.73 - 7.60i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (0.260 + 0.327i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-2.96 + 13.0i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 3.56T + 59T^{2} \)
61 \( 1 + (6.49 - 3.12i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-5.18 + 6.49i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (6.10 + 7.65i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.0675 - 0.296i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-2.49 + 3.13i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-3.88 - 1.87i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (3.11 - 13.6i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-3.42 - 1.65i)T + (60.4 + 75.8i)T^{2} \)
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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

−13.27597957375975188454368204586, −12.58580175549821068866002180256, −11.80344916604537076557649650173, −10.82495597698401342954435669787, −9.576303913789971066521795123626, −8.950182616010774040382222507793, −6.55411598188225544733346104699, −4.89761316962431354705074946686, −3.87265443812739292850745047022, −1.62846935687453105587633689092, 3.66412296839382104671714134181, 5.78141329376088827418286144292, 6.23529172208404526055573555449, 7.37285359672645065145755349783, 8.502738285757366532595159419896, 10.28336960642424582854238264586, 11.22897824006293087143501189909, 12.91206598511983299991666427368, 13.71494669460510920734629897820, 14.71518092660120876197321972286

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