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Properties

Label 2-13e2-13.10-c3-0-8
Degree $2$
Conductor $169$
Sign $0.252 - 0.967i$
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  + (−1.5 + 0.866i)2-s + (−1 − 1.73i)3-s + (−2.5 + 4.33i)4-s − 1.73i·5-s + (3 + 1.73i)6-s + (12 + 6.92i)7-s − 22.5i·8-s + (11.5 − 19.9i)9-s + (1.49 + 2.59i)10-s + (−12 + 6.92i)11-s + 10·12-s − 24·14-s + (−2.99 + 1.73i)15-s + (−0.500 − 0.866i)16-s + (−58.5 + 101. i)17-s + 39.8i·18-s + ⋯
L(s)  = 1  + (−0.530 + 0.306i)2-s + (−0.192 − 0.333i)3-s + (−0.312 + 0.541i)4-s − 0.154i·5-s + (0.204 + 0.117i)6-s + (0.647 + 0.374i)7-s − 0.995i·8-s + (0.425 − 0.737i)9-s + (0.0474 + 0.0821i)10-s + (−0.328 + 0.189i)11-s + 0.240·12-s − 0.458·14-s + (−0.0516 + 0.0298i)15-s + (−0.00781 − 0.0135i)16-s + (−0.834 + 1.44i)17-s + 0.521i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.832632 + 0.643152i\)
\(L(\frac12)\) \(\approx\) \(0.832632 + 0.643152i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 - 0.866i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 1.73iT - 125T^{2} \)
7 \( 1 + (-12 - 6.92i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (12 - 6.92i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (58.5 - 101. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-99 - 57.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39 - 67.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-70.5 - 122. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 155. iT - 2.97e4T^{2} \)
37 \( 1 + (-124.5 + 71.8i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (235.5 - 135. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (52 - 90.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 301. iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (-246 - 142. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.5 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-681 + 393. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (915 + 528. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 458. iT - 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 789. iT - 5.71e5T^{2} \)
89 \( 1 + (-846 + 488. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (174 + 100. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.56419516355156520454756162022, −11.71944963089547327153897892355, −10.39028590236463935327681274345, −9.240192942577938792149626266221, −8.415194175077872610888586081850, −7.45683561700817193666835789616, −6.40917635934714065641302057123, −4.88911036549834203794276131516, −3.48454413231220453678319345641, −1.31414323649380026611142289833, 0.69775681349343177399839854730, 2.43282159643065322809981699102, 4.62366077445546092077662277598, 5.24446126832002427994338826621, 6.99539880623038828219996358249, 8.136885324681471532042815750738, 9.274941552613008166721962211635, 10.16052625422979483368065081463, 11.01728697458626857279640365103, 11.59275452915860328983811817983

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