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Properties

Label 2-29-29.4-c1-0-0
Degree $2$
Conductor $29$
Sign $0.980 - 0.195i$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
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  + (−0.536 − 0.122i)2-s + (0.855 + 1.77i)3-s + (−1.52 − 0.736i)4-s + (0.610 − 2.67i)5-s + (−0.241 − 1.05i)6-s + (−4.03 + 1.94i)7-s + (1.58 + 1.26i)8-s + (−0.556 + 0.697i)9-s + (−0.654 + 1.35i)10-s + (1.16 − 0.929i)11-s − 3.34i·12-s + (0.906 + 1.13i)13-s + (2.39 − 0.547i)14-s + (5.27 − 1.20i)15-s + (1.41 + 1.78i)16-s + 2.07i·17-s + ⋯
L(s)  = 1  + (−0.379 − 0.0865i)2-s + (0.494 + 1.02i)3-s + (−0.764 − 0.368i)4-s + (0.273 − 1.19i)5-s + (−0.0985 − 0.431i)6-s + (−1.52 + 0.733i)7-s + (0.562 + 0.448i)8-s + (−0.185 + 0.232i)9-s + (−0.207 + 0.429i)10-s + (0.351 − 0.280i)11-s − 0.966i·12-s + (0.251 + 0.315i)13-s + (0.640 − 0.146i)14-s + (1.36 − 0.311i)15-s + (0.354 + 0.445i)16-s + 0.502i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577238 + 0.0569274i\)
\(L(\frac12)\) \(\approx\) \(0.577238 + 0.0569274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.78 + 3.83i)T \)
good2 \( 1 + (0.536 + 0.122i)T + (1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.855 - 1.77i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (-0.610 + 2.67i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (4.03 - 1.94i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-1.16 + 0.929i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.906 - 1.13i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 2.07iT - 17T^{2} \)
19 \( 1 + (0.236 - 0.490i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (0.382 + 1.67i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (4.54 + 1.03i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (2.37 + 1.89i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 + 0.595iT - 41T^{2} \)
43 \( 1 + (-0.321 + 0.0733i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-8.97 + 7.15i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (2.28 - 10.0i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + 4.03T + 59T^{2} \)
61 \( 1 + (-5.18 - 10.7i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (5.20 - 6.52i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (5.78 + 7.25i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.37 - 0.769i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (-6.50 - 5.19i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (0.955 + 0.460i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-4.68 - 1.06i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-2.09 + 4.35i)T + (-60.4 - 75.8i)T^{2} \)
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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

−16.94393094706047321622056833673, −16.15547853975778271345648594908, −15.00204158920203688087174034810, −13.51697837277618606582554393422, −12.49016537513863759708879377597, −10.24074991179619322850723908248, −9.157158320988940573081380684738, −8.889036090441205417615811614385, −5.73960901976586531278033097929, −4.03063395235420135060249814795, 3.33608310731647762704031487964, 6.70726854227879352784738010540, 7.50772521545945450966998841694, 9.310468364108848429253135013693, 10.46992782139703086448990647580, 12.66740352058450840295971734300, 13.45265381453446472231804151977, 14.32112164011568789258405910800, 16.16399155600815304047166333441, 17.48673192158270597530198471863

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