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Properties

Label 2-29e2-29.9-c1-0-34
Degree $2$
Conductor $841$
Sign $0.974 + 0.225i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
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  + (0.268 + 0.556i)2-s + (−0.483 + 0.385i)3-s + (1.00 − 1.26i)4-s + (3.47 − 1.67i)5-s + (−0.344 − 0.165i)6-s + (−1.39 − 1.74i)7-s + (2.18 + 0.497i)8-s + (−0.582 + 2.55i)9-s + (1.86 + 1.48i)10-s + (1.34 − 0.307i)11-s + i·12-s + (−0.0525 − 0.230i)13-s + (0.599 − 1.24i)14-s + (−1.03 + 2.14i)15-s + (−0.412 − 1.80i)16-s + 4.38i·17-s + ⋯
L(s)  = 1  + (0.189 + 0.393i)2-s + (−0.278 + 0.222i)3-s + (0.504 − 0.632i)4-s + (1.55 − 0.747i)5-s + (−0.140 − 0.0676i)6-s + (−0.526 − 0.660i)7-s + (0.770 + 0.175i)8-s + (−0.194 + 0.850i)9-s + (0.588 + 0.469i)10-s + (0.406 − 0.0927i)11-s + 0.288i·12-s + (−0.0145 − 0.0638i)13-s + (0.160 − 0.332i)14-s + (−0.266 + 0.554i)15-s + (−0.103 − 0.451i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.974 + 0.225i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29435 - 0.261550i\)
\(L(\frac12)\) \(\approx\) \(2.29435 - 0.261550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-0.268 - 0.556i)T + (-1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.483 - 0.385i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (-3.47 + 1.67i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.39 + 1.74i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-1.34 + 0.307i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.0525 + 0.230i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 4.38iT - 17T^{2} \)
19 \( 1 + (-3.79 - 3.02i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-1.11 - 0.536i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (4.37 + 9.09i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (4.59 + 1.04i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 - 3.85iT - 41T^{2} \)
43 \( 1 + (-3.13 + 6.51i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.82 - 1.55i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (-1.80 + 0.867i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (0.483 - 0.385i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (0.339 - 1.48i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-2.33 - 10.2i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.94 - 12.3i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (5.93 + 1.35i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (6.20 - 7.77i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-2.04 - 4.24i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (-2.78 - 2.22i)T + (21.5 + 94.5i)T^{2} \)
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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

−10.03421260845797390667776207487, −9.695562789867447831147096638734, −8.517814706729050892492237645071, −7.41958725448610246561045206811, −6.42746754220080940431438546614, −5.67191427353974833614889440510, −5.27562976120342545025929740840, −4.02807800356887535552407741813, −2.24948435261610742183453717790, −1.28630566606136308687533845422, 1.60578060547623550960346853358, 2.81093300795874305160856761746, 3.28019624937593781073364121165, 5.05787770289355342679269199933, 6.02182471024711003030988350320, 6.76357733486929382663681140612, 7.22585570419334655036348968948, 8.932150416736550701002117879392, 9.381434940019659459543516312617, 10.27749940651262789657600911089

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