Properties

Label 1-89-89.25-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.478 - 0.878i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.841 − 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.654 + 0.755i)5-s + (0.959 + 0.281i)6-s + (0.654 − 0.755i)7-s + (−0.654 + 0.755i)8-s + (0.415 + 0.909i)9-s + (0.415 − 0.909i)10-s + (−0.654 − 0.755i)11-s − 12-s + (−0.841 − 0.540i)13-s + (−0.415 + 0.909i)14-s + (0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.959 − 0.281i)17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−0.841 − 0.540i)3-s + (0.841 − 0.540i)4-s + (−0.654 + 0.755i)5-s + (0.959 + 0.281i)6-s + (0.654 − 0.755i)7-s + (−0.654 + 0.755i)8-s + (0.415 + 0.909i)9-s + (0.415 − 0.909i)10-s + (−0.654 − 0.755i)11-s − 12-s + (−0.841 − 0.540i)13-s + (−0.415 + 0.909i)14-s + (0.959 − 0.281i)15-s + (0.415 − 0.909i)16-s + (−0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.478 - 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1329091484 - 0.2237411746i\)
\(L(\frac12)\) \(\approx\) \(0.1329091484 - 0.2237411746i\)
\(L(1)\) \(\approx\) \(0.3996630700 - 0.1008928913i\)
\(L(1)\) \(\approx\) \(0.3996630700 - 0.1008928913i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.959 + 0.281i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.841 + 0.540i)T \)
61 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.654 + 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.891488280025749242909371021268, −29.242177556406227944802619274978, −28.6045028664370583939977601665, −27.655491234073825961372986182778, −27.19540722603618946903054737045, −25.87015139551686923453743777151, −24.46289221882647767819537780946, −23.69639104750968080586794517703, −22.08762514003971498643087735542, −21.1116960670007906885270555459, −20.2569523513733973767899001110, −18.92466515339881297431678283968, −17.76269436624046836248773864361, −16.98356157455331388453409122969, −15.80313100231890643820867211691, −15.144674851267925079505372103972, −12.47959047764968885702134283339, −11.94641019643040354138472407400, −10.82454723560573874892930070220, −9.57396731685662385958811446698, −8.49068350097684097772540015153, −7.18487980626403111316187457041, −5.42711441873770186112445306047, −4.139364646028902238450654760967, −1.90484667463668459004280515827, 0.39807449277491659957738578097, 2.470161296038023721717249458, 4.878378865833615424299531133620, 6.50293650678459969147717582951, 7.38259682897723558705925400763, 8.308436255565666244337164405115, 10.48391019205323486182204301308, 10.895674468027275045621424002174, 12.02467636562278261776366939715, 13.78148200136776472443263512416, 15.20068311716024892408386921685, 16.25959316096904058330827592023, 17.451420732618999777925682831394, 18.11645145755625232273240881921, 19.19345606711621065465424235014, 20.09224567961651929707124018274, 21.763438774192647685973675552034, 23.11155688041385917372694083995, 23.96054087722006070091181134891, 24.69614164611962223449108642464, 26.404147742401380137800534635045, 26.95460808467058785846632101317, 27.939637524180438631833260763287, 29.13579135666925530337462128545, 29.90747893302930717716682481021

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