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Properties

Label 2-26-13.7-c2-0-1
Degree $2$
Conductor $26$
Sign $0.923 + 0.383i$
Analytic cond. $0.708448$
Root an. cond. $0.841693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (1.92 − 3.33i)3-s + (1.73 − i)4-s + (3.77 + 3.77i)5-s + (−1.40 + 5.25i)6-s + (−9.91 − 2.65i)7-s + (−1.99 + 2i)8-s + (−2.90 − 5.02i)9-s + (−6.53 − 3.77i)10-s + (2.71 + 10.1i)11-s − 7.69i·12-s + (−8.18 + 10.0i)13-s + 14.5·14-s + (19.8 − 5.31i)15-s + (1.99 − 3.46i)16-s + (4.23 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.641 − 1.11i)3-s + (0.433 − 0.250i)4-s + (0.754 + 0.754i)5-s + (−0.234 + 0.875i)6-s + (−1.41 − 0.379i)7-s + (−0.249 + 0.250i)8-s + (−0.322 − 0.558i)9-s + (−0.653 − 0.377i)10-s + (0.246 + 0.919i)11-s − 0.641i·12-s + (−0.629 + 0.776i)13-s + 1.03·14-s + (1.32 − 0.354i)15-s + (0.124 − 0.216i)16-s + (0.249 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(0.708448\)
Root analytic conductor: \(0.841693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :1),\ 0.923 + 0.383i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.822404 - 0.163920i\)
\(L(\frac12)\) \(\approx\) \(0.822404 - 0.163920i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
13 \( 1 + (8.18 - 10.0i)T \)
good3 \( 1 + (-1.92 + 3.33i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.77 - 3.77i)T + 25iT^{2} \)
7 \( 1 + (9.91 + 2.65i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-2.71 - 10.1i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-4.23 + 2.44i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.83 + 25.5i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (17.2 + 9.97i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (7.15 - 12.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (-19.0 - 19.0i)T + 961iT^{2} \)
37 \( 1 + (15.7 + 58.6i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (4.83 - 1.29i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (10.3 - 5.98i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (7.59 - 7.59i)T - 2.20e3iT^{2} \)
53 \( 1 - 77.0T + 2.80e3T^{2} \)
59 \( 1 + (60.7 + 16.2i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-28.1 - 48.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.90 + 1.58i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (14.7 - 55.2i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-12.7 + 12.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 7.98T + 6.24e3T^{2} \)
83 \( 1 + (35.8 + 35.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (-20.9 - 78.2i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (14.1 - 52.9i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−17.53031204569929958531291977081, −16.13559619258165169677728024550, −14.50658791491638921249273604934, −13.54321777166127587621931512555, −12.30483861467732624535266742285, −10.23735311132700076069389175781, −9.228267890320454449470469874272, −7.19686918574269283431349104819, −6.64726784752073974216223813522, −2.45694893223072642811104559116, 3.32403950724458260739668169059, 5.86126712498175236756529226101, 8.414200846963753870410423786743, 9.643558416114212156081472626078, 10.04056676853149559398374877139, 12.23834017611332649565484381074, 13.59284591180924489086241131861, 15.23668404892895756445575741205, 16.26704206035522076080843420517, 17.00013548070062732803895849153

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