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Properties

Label 2-7e2-49.36-c1-0-0
Degree $2$
Conductor $49$
Sign $0.442 - 0.896i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
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  + (0.367 + 1.61i)2-s + (0.290 + 0.364i)3-s + (−0.658 + 0.317i)4-s + (−2.42 − 3.04i)5-s + (−0.480 + 0.602i)6-s + (−1.62 + 2.08i)7-s + (1.30 + 1.63i)8-s + (0.619 − 2.71i)9-s + (4.00 − 5.02i)10-s + (−0.173 − 0.759i)11-s + (−0.306 − 0.147i)12-s + (0.647 + 2.83i)13-s + (−3.96 − 1.85i)14-s + (0.403 − 1.76i)15-s + (−3.07 + 3.85i)16-s + (−2.90 − 1.39i)17-s + ⋯
L(s)  = 1  + (0.260 + 1.13i)2-s + (0.167 + 0.210i)3-s + (−0.329 + 0.158i)4-s + (−1.08 − 1.35i)5-s + (−0.196 + 0.245i)6-s + (−0.614 + 0.788i)7-s + (0.462 + 0.579i)8-s + (0.206 − 0.904i)9-s + (1.26 − 1.58i)10-s + (−0.0522 − 0.229i)11-s + (−0.0885 − 0.0426i)12-s + (0.179 + 0.787i)13-s + (−1.05 − 0.494i)14-s + (0.104 − 0.456i)15-s + (−0.768 + 0.963i)16-s + (−0.703 − 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.442 - 0.896i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.442 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750077 + 0.466502i\)
\(L(\frac12)\) \(\approx\) \(0.750077 + 0.466502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.62 - 2.08i)T \)
good2 \( 1 + (-0.367 - 1.61i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.290 - 0.364i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.42 + 3.04i)T + (-1.11 + 4.87i)T^{2} \)
11 \( 1 + (0.173 + 0.759i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.647 - 2.83i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (2.90 + 1.39i)T + (10.5 + 13.2i)T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 + (2.84 - 1.37i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.0505 + 0.0243i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 0.919T + 31T^{2} \)
37 \( 1 + (4.59 + 2.21i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (0.649 + 0.814i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-6.86 + 8.61i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.29 - 10.0i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-9.54 + 4.59i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-1.68 + 2.10i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.73 - 0.836i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 - 0.344T + 67T^{2} \)
71 \( 1 + (5.76 - 2.77i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.483 - 2.11i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 + (0.144 - 0.632i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.86 - 8.17i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + 9.01T + 97T^{2} \)
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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

−15.91209277380781898075543257295, −15.21341887440188450415098218919, −13.79230410508697590890915144598, −12.43551620112880000658877145003, −11.57791149600485616925067782507, −9.275562062089993295067668732489, −8.498324390314625054262509027897, −7.05352430831926691536251050013, −5.57194209151323007405562081865, −4.12185262540218666767795285872, 2.84183454217046821643758793370, 4.04278073394328058525189748727, 6.91994243624816923514167276623, 7.76897453866288908866409469874, 10.23880489809639528155236839573, 10.72957823604910475056809270274, 11.80726778301538017538605468718, 13.05352691322224883053400481752, 13.97618384070318994784898209539, 15.43530340407229908486387141993

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