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Properties

Label 2-13e2-13.4-c3-0-15
Degree $2$
Conductor $169$
Sign $-0.802 - 0.596i$
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  + (4.17 + 2.41i)2-s + (−2.22 + 3.84i)3-s + (7.64 + 13.2i)4-s + 12.7i·5-s + (−18.5 + 10.7i)6-s + (22.6 − 13.0i)7-s + 35.1i·8-s + (3.62 + 6.28i)9-s + (−30.8 + 53.3i)10-s + (−36.6 − 21.1i)11-s − 67.9·12-s + 126.·14-s + (−49.1 − 28.3i)15-s + (−23.6 + 41.0i)16-s + (−13.6 − 23.6i)17-s + 35.0i·18-s + ⋯
L(s)  = 1  + (1.47 + 0.853i)2-s + (−0.427 + 0.740i)3-s + (0.955 + 1.65i)4-s + 1.14i·5-s + (−1.26 + 0.729i)6-s + (1.22 − 0.706i)7-s + 1.55i·8-s + (0.134 + 0.232i)9-s + (−0.974 + 1.68i)10-s + (−1.00 − 0.580i)11-s − 1.63·12-s + 2.41·14-s + (−0.845 − 0.488i)15-s + (−0.370 + 0.641i)16-s + (−0.194 − 0.337i)17-s + 0.458i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.802 - 0.596i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.802 - 0.596i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11307 + 3.36596i\)
\(L(\frac12)\) \(\approx\) \(1.11307 + 3.36596i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.17 - 2.41i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (2.22 - 3.84i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 - 12.7iT - 125T^{2} \)
7 \( 1 + (-22.6 + 13.0i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (36.6 + 21.1i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (13.6 + 23.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-11.3 + 6.55i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (14.3 - 24.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.8 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 56.0iT - 2.97e4T^{2} \)
37 \( 1 + (-271. - 156. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-305. - 176. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (160. + 277. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 339. iT - 1.03e5T^{2} \)
53 \( 1 - 349.T + 1.48e5T^{2} \)
59 \( 1 + (-224. + 129. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (325. + 563. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (775. + 447. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (642. - 370. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 820. iT - 3.89e5T^{2} \)
79 \( 1 + 199.T + 4.93e5T^{2} \)
83 \( 1 + 541. iT - 5.71e5T^{2} \)
89 \( 1 + (329. + 190. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.23e3 + 715. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.16548681020884851950699064379, −11.58020688475279259336441361919, −10.99849327826906488637257920122, −10.15403653474060986310130960291, −7.967505270525076619313299982911, −7.32111828154309161226495988174, −6.05236845569239261889262419055, −5.00077421457666629808981209366, −4.23223274660061885557764930844, −2.83616970330542147195694925195, 1.23263124285666686673551752770, 2.32157227741353093952358348704, 4.33617977047527378671994203753, 5.14477229918339375797753030730, 5.95967196642390449128873427592, 7.61391761350824031432677045353, 8.850927913161686439447284852594, 10.41134086475580474172369076468, 11.52531030995581944767255882848, 12.14807876263335804297407474279

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