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Properties

Label 1-7e2-49.47-r1-0-0
Degree $1$
Conductor $49$
Sign $-0.335 + 0.942i$
Analytic cond. $5.26578$
Root an. cond. $5.26578$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 + 0.563i)2-s + (0.733 + 0.680i)3-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.365 + 0.930i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.733 + 0.680i)3-s + (0.365 + 0.930i)4-s + (−0.955 + 0.294i)5-s + (0.222 + 0.974i)6-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.365 + 0.930i)12-s + (0.900 − 0.433i)13-s + (−0.900 − 0.433i)15-s + (−0.733 + 0.680i)16-s + (0.988 − 0.149i)17-s + (−0.5 + 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.335 + 0.942i$
Analytic conductor: \(5.26578\)
Root analytic conductor: \(5.26578\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (1:\ ),\ -0.335 + 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.498999096 + 2.124645333i\)
\(L(\frac12)\) \(\approx\) \(1.498999096 + 2.124645333i\)
\(L(1)\) \(\approx\) \(1.473372345 + 1.112401868i\)
\(L(1)\) \(\approx\) \(1.473372345 + 1.112401868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.826 + 0.563i)T \)
3 \( 1 + (0.733 + 0.680i)T \)
5 \( 1 + (-0.955 + 0.294i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.900 - 0.433i)T \)
17 \( 1 + (0.988 - 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (-0.0747 - 0.997i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.69798329279304188998156144467, −31.82537033846294977852666001784, −30.685802120683858567535638836663, −30.3852677361861383787510658199, −28.73191816536187377745403392158, −27.77003107515329327763794085472, −26.06016450251027367410465030427, −24.775043488472913260679752774825, −23.66541143591224385602621933426, −23.00009051118129372939806813193, −21.18373820649440034910281697581, −20.14559073566552495715414754522, −19.40090558638187859935299079778, −18.18761983426815464266760799256, −15.91047852037646580972278857270, −14.84756356485968910549587513872, −13.639051779316848006435831944223, −12.442590513979978580244676949850, −11.57786132496090268158178186890, −9.659730544046656397313371217553, −7.987515825683835703241064665538, −6.56138054328750788249965052878, −4.47699108332138727294038574289, −3.14972440516770151768680779788, −1.31805734815533345136931076990, 3.15278531153968489416904529165, 3.98342644644437901813898265149, 5.73096614288464946368131664696, 7.66393434606852569969255157164, 8.53062091734115035492642381435, 10.654137965038325367938804128961, 12.016913004479618822132075933332, 13.701642444135043655618678434412, 14.610495709518129387563305038197, 15.83101295944673041027244459046, 16.42877469603293426372974434445, 18.62801505199003145717336115644, 20.06835389341915030781157980170, 21.0940903511795132937946161221, 22.28544245282150624577941207741, 23.33012668118290049068770380206, 24.60452839319184072577421403194, 25.78478847099131998767657074501, 26.75115188076670870015789722368, 27.72922373450112081917194256891, 29.83858962588714893611476764309, 30.829458149806561217765099191529, 31.760501601904315169652906955392, 32.50086770484082794390270381624, 33.72701521486366395295044721290

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