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Properties

Label 2-13e2-13.3-c3-0-11
Degree $2$
Conductor $169$
Sign $-0.379 + 0.925i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
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  + (−2.42 − 4.19i)2-s + (3.09 + 5.36i)3-s + (−7.71 + 13.3i)4-s − 15.2·5-s + (14.9 − 25.9i)6-s + (2.15 − 3.73i)7-s + 35.9·8-s + (−5.69 + 9.87i)9-s + (36.8 + 63.8i)10-s + (12.2 + 21.2i)11-s − 95.5·12-s − 20.8·14-s + (−47.2 − 81.7i)15-s + (−25.2 − 43.8i)16-s + (63.5 − 110. i)17-s + 55.1·18-s + ⋯
L(s)  = 1  + (−0.855 − 1.48i)2-s + (0.596 + 1.03i)3-s + (−0.964 + 1.67i)4-s − 1.36·5-s + (1.02 − 1.76i)6-s + (0.116 − 0.201i)7-s + 1.58·8-s + (−0.211 + 0.365i)9-s + (1.16 + 2.02i)10-s + (0.336 + 0.583i)11-s − 2.29·12-s − 0.398·14-s + (−0.812 − 1.40i)15-s + (−0.395 − 0.684i)16-s + (0.907 − 1.57i)17-s + 0.722·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.379 + 0.925i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.379 + 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.460430 - 0.686392i\)
\(L(\frac12)\) \(\approx\) \(0.460430 - 0.686392i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (2.42 + 4.19i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-3.09 - 5.36i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 15.2T + 125T^{2} \)
7 \( 1 + (-2.15 + 3.73i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-12.2 - 21.2i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-63.5 + 110. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-25.8 + 44.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (43.6 + 75.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (112. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 108.T + 2.97e4T^{2} \)
37 \( 1 + (57.8 + 100. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-95.9 - 166. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-61.6 + 106. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 36.7T + 1.03e5T^{2} \)
53 \( 1 - 119.T + 1.48e5T^{2} \)
59 \( 1 + (-402. + 696. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (339. - 587. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-43.7 - 75.7i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-490. + 849. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 263.T + 3.89e5T^{2} \)
79 \( 1 - 321.T + 4.93e5T^{2} \)
83 \( 1 + 1.04e3T + 5.71e5T^{2} \)
89 \( 1 + (-172. - 298. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-241. + 417. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.76958006993685481520946760073, −10.97998333272784244828141381076, −9.879643376479159696277490853501, −9.362029829308976452577904333094, −8.284898793064608085783597604321, −7.38638567744873162212180896730, −4.55904323858509427858968893358, −3.78646369937656865081338091084, −2.74896085645125159519046523020, −0.57477301280130435426359043692, 1.20759246794816029454293042948, 3.66536740685364523214949718793, 5.57802633957891116570173379045, 6.75704641023544921704236413009, 7.71452701347290749586588530133, 8.117436499771690708578355111234, 8.883336663076914095369207751322, 10.35926921941374796958110868037, 11.78074498112670909471469979560, 12.71410652145989058789147338285

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