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Properties

Label 2-13e2-13.10-c1-0-0
Degree $2$
Conductor $169$
Sign $-0.425 - 0.905i$
Analytic cond. $1.34947$
Root an. cond. $1.16166$
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  + (−0.694 + 0.400i)2-s + (1.12 + 1.94i)3-s + (−0.678 + 1.17i)4-s − 0.246i·5-s + (−1.56 − 0.900i)6-s + (2.04 + 1.17i)7-s − 2.69i·8-s + (−1.02 + 1.77i)9-s + (0.0990 + 0.171i)10-s + (−3.67 + 2.12i)11-s − 3.04·12-s − 1.89·14-s + (0.480 − 0.277i)15-s + (−0.277 − 0.480i)16-s + (1.07 − 1.86i)17-s − 1.64i·18-s + ⋯
L(s)  = 1  + (−0.491 + 0.283i)2-s + (0.648 + 1.12i)3-s + (−0.339 + 0.587i)4-s − 0.110i·5-s + (−0.637 − 0.367i)6-s + (0.771 + 0.445i)7-s − 0.951i·8-s + (−0.341 + 0.591i)9-s + (0.0313 + 0.0542i)10-s + (−1.10 + 0.640i)11-s − 0.880·12-s − 0.505·14-s + (0.124 − 0.0716i)15-s + (−0.0693 − 0.120i)16-s + (0.261 − 0.453i)17-s − 0.387i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.425 - 0.905i$
Analytic conductor: \(1.34947\)
Root analytic conductor: \(1.16166\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1/2),\ -0.425 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545818 + 0.859654i\)
\(L(\frac12)\) \(\approx\) \(0.545818 + 0.859654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.694 - 0.400i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.12 - 1.94i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.246iT - 5T^{2} \)
7 \( 1 + (-2.04 - 1.17i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.07 + 1.86i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0763 + 0.0440i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.746 - 1.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.31 + 4.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.63iT - 31T^{2} \)
37 \( 1 + (-4.92 + 2.84i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.0 + 5.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.147 - 0.256i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.35iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-5.87 - 3.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.73 - 3.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.65 - 3.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.50 + 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.73iT - 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 + 1.60iT - 83T^{2} \)
89 \( 1 + (2.49 - 1.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.97 + 4.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.09966624080924495811494180456, −12.17338466995095057719303086865, −10.79966153109508107650676499502, −9.810214338901377848460215350114, −9.017310662846521310904093477500, −8.211297077858277505294375508018, −7.27673323383809519736967793268, −5.20329357367445745001922613946, −4.25962704848026034773377605972, −2.83908377806447979188323190513, 1.21451106900017660873886776078, 2.64709195197643084460540848858, 4.80587546309102196319333644310, 6.16698918106495844932933058100, 7.72922351196120073730191895364, 8.140649870694542806693601926026, 9.297590387297452233578553450199, 10.62800815033460750488606271854, 11.17035449771696440833282600054, 12.73460244218095652448515900584

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