Properties

Degree $1$
Conductor $31$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s,χ)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s − 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s − 22-s − 23-s − 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{31} (30, \cdot )$
Sato-Tate group: $\mu(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(2.208631876\)
\(L(\frac12,\chi)\) \(\approx\) \(2.208631876\)
\(L(\chi,1)\) \(\approx\) \(1.692740092\)
\(L(1,\chi)\) \(\approx\) \(1.692740092\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−36.67685000147278202904180240027, −34.73525141826695901662244337819, −33.72763908065133135528759026996, −33.21474384193483545413746361975, −31.63934929805853763413028640302, −30.22023902961814429222295335058, −29.23477470645741101301735856374, −28.3374356036886045704236376649, −26.4601535274486392459102175141, −24.59995978410064295214080086541, −24.011330598375603608659460143202, −22.42070185559805391300678577861, −21.6070391912365273178177076636, −20.49918501249093324848331385672, −18.1687248306068221894848784442, −17.09514996166819244174822299264, −15.60301752682389109688066981158, −14.05885519779649141901602534311, −12.77142050335587766094724998350, −11.39258383036663027678688446981, −10.18385049058425751554958684335, −7.33591057329007776327141175518, −5.68644602105895428746100540523, −4.78968470194607603822353764529, −2.03498242395270329177915345847, 2.03498242395270329177915345847, 4.78968470194607603822353764529, 5.68644602105895428746100540523, 7.33591057329007776327141175518, 10.18385049058425751554958684335, 11.39258383036663027678688446981, 12.77142050335587766094724998350, 14.05885519779649141901602534311, 15.60301752682389109688066981158, 17.09514996166819244174822299264, 18.1687248306068221894848784442, 20.49918501249093324848331385672, 21.6070391912365273178177076636, 22.42070185559805391300678577861, 24.011330598375603608659460143202, 24.59995978410064295214080086541, 26.4601535274486392459102175141, 28.3374356036886045704236376649, 29.23477470645741101301735856374, 30.22023902961814429222295335058, 31.63934929805853763413028640302, 33.21474384193483545413746361975, 33.72763908065133135528759026996, 34.73525141826695901662244337819, 36.67685000147278202904180240027

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